Celebratio Mathematica — Rosenblatt

[1] M. Rosenblatt: On distributions of certain Wiener functionals. Ph.D. thesis, Cornell University, 1949. Advised by M. Kac. MR 2937915 phdthesis

[2] M. Rosenblatt: “On a class of Markov processes,” Trans. Am. Math. Soc. 71 : 1 (1951), pp. 120–135. MR 43406 Zbl 0045.07703 article

[3] M. Rosenblatt: “On the oscillation of sums of random variables,” Trans. Am. Math. Soc. 72 : 1 (1952), pp. 165–178. MR 45326 Zbl 0046.35202 article

Let \( X \) be a random variable with distribution function \( F(x) \) and characteristic function \[ \phi(z) = \int_{-\infty}^{\infty}e^{izx}dF(x) .\] The sequence of partial sums \( \{S_n\} \) will be said to be generated by \( X \) if \[ S_n = \sum_{j=1}^n X_j ,\] where \( X_1,\dots \), \( X_n,\dots \) are independent, identically distributed random variables with distribution function \( F(x) \).

Let the abbreviations i.o. and f.o. denote the phrases “infinitely often” and “finitely often,” respectively. The sequence \( \{S_n\} \) generated by the random variable \( X \) is said to oscillate if \[ P\{S_n > 0 \ \mathrm{i.o.} \} = P\{S_n\leq 0 \ \mathrm{i.o.} \} = 1. \]

A sufficient condition for oscillation of the sequence \( S_n \), convenient for the application of results concerning partial limit laws of normed sums of independent and identically distributed random variables, will be obtained. When \( E(|X|) < \infty \), the problem considered will be shown to be equivalent to a problem dealt with by K. L. Chung and W. H. J. Fuchs [1951].

The necessary and sufficient condition for the oscillation of the sequence \( \{S_n\} \) generated by \( X \) will be obtained. The necessary and sufficient condition is used to obtain a class of random variables each of which generates sums \( S_n \) which satisfy \begin{align*} P\bigr\{\lim_{n\to\infty} |S_n| = \infty\bigl\} &= P\bigl\{\liminf_{n\to\infty} S_n = -\infty\bigr\} \\ &= P\bigl\{\limsup_{n\to\infty} S_n = \infty\bigr\} = 1 \end{align*} and \[ \lim_{n\to\infty} P\{S_n > 0\} - 0 \] simultaneously.

[4] M. Rosenblatt: “The behavior at zero of the characteristic function of a random variable,” Proc. Am. Math. Soc. 3 : 3 (1952), pp. 498–504. MR 47957 Zbl 0047.12206 article

[5] U. Grenander and M. Rosenblatt: “On spectral analysis of stationary time series,” Proc. Natl. Acad. Sci. U.S.A. 38 : 6 (June 1952), pp. 519–521. MR 48737 Zbl 0047.12503 article

[6] M. Rosenblatt: “Remarks on a multivariate transformation,” Ann. Math. Stat. 23 : 3 (1952), pp. 470–472. MR 49525 Zbl 0047.13104 article

[7] M. Rosenblatt: “Limit theorems associated with variants of the von Mises statistic,” Ann. Math. Stat. 23 : 4 (1952), pp. 617–623. MR 52732 Zbl 0048.36003 article

A multidimensional analogue of the von Mises statistic is considered for the case of sampling from a multidimensional uniform distribution. The limiting distribution of the statistic is shown to be that of a weighted sum of independent chi-square random variables with one degree of freedom. The weights are the eigenvalues of a positive definite symmetric function. A modified statistic of the von Mises type useful in setting up a two sample test is shown to have the same limiting distribution under the null hypothesis (both samples come from the same population with a continuous distribution function) as that of the one-dimensional von Mises statistic. We call the statistics mentioned above von Mises statistics because they are modifications of the \( \omega^2 \) criterion considered by von Mises [Smirnoff 1936]. The paper makes use of elements of the theory of stochastic processes.

[8] J. L. Hodges, Jr. and M. Rosenblatt: “Recurrence-time moments in random walks,” Pac. J. Math. 3 : 1 (1953), pp. 127–136. MR 54190 Zbl 0050.35402 article

Consider an irreducible time-homogeneous Markov chain with discrete time. The recurrence-time moments of the states of such stochastic processes are studied. We point out that if the recurrence time of one state has its first \( k \) moments finite, then the recurrence times of all the other states have their first \( k \) moments finite. We then specialize and investigate the recurrence time moments of random walks. The main result of the paper consists of exhibiting random walks whose first \( k-1 \) recurrence-time moments exist and whose higher moments are infinite, for \( k = 1 \), \( 2,\dots \). A comparison theorem is derived that permits the moment properties of recurrence times of a large class of random walks to be determined.

[9] K. A. Brownlee, J. L. Hodges, Jr., and M. Rosenblatt: “The up-and-down method with small samples,” J. Am. Stat. Assoc. 48 : 262 (1953), pp. 262–277. MR 55644 Zbl 0050.36002 article

Kenneth Alexander Brownlee	Related
Joseph Lawson Hodges, Jr.	Related

[10] U. Grenander and M. Rosenblatt: “Statistical spectral analysis of time series arising from stationary stochastic processes,” Ann. Math. Stat. 24 : 4 (December 1953), pp. 537–558. MR 58901 Zbl 0053.41005 article

[11] U. Grenander and M. Rosenblatt: “Comments on statistical spectral analysis,” Skand. Aktuarietidskr. 36 : Suppl. 1 (1953), pp. 182–202. MR 60796 Zbl 0053.41101 article

One of the objects of this paper is to give a heuristic derivation of the main result in [Grenander and Rosenblatt 1953] and to camment on its practical application in time series analysis. The rigorous proofs in [Grenander and Rosenblatt 1953] were rather detailed and laborious. We feel that this paper will be useful to the reader whose main interest is not in mathematical niceties. Time series analysis is of great practical importance and has grown rapidly in the past few years. Much work is yet to be done. The comments in this paper should not he interpreted as recommendations for optimal procedures. Tables and graphs of relevant limiting distribution functions are given. Although this paper deals with the distribution theory of a class of estimates of the spectral distribution function of a real stationary time series, we shall first consider some nonstatistical aspects of time series analysis.

[12] U. Grenander and M. Rosenblatt: “An extension of a theorem of G. Szegő and its application to the study of stochastic processes,” Trans. Am. Math. Soc. 76 : 1 (January 1954), pp. 112–126. MR 58902 Zbl 0059.11804 article

In this paper we study minimum problems associated with quadratic forms \[ Q_n = c^{\prime}M^{(n)}c \] where \( c \) is a column vector with components \( c_0 \), \( c_1, \dots \), \( c_n \) and \( M^{(n)} \) is a Hermitian matrix with the elements \[ m_{p,q}^{(n)} = \int_{-\pi}^{\pi} e^{i(p-q)\lambda}f(\lambda)\,d\lambda, \quad p,q = 0,1,\dots,n. \] We denote the conjugate of the transpose of a matrix \( A \) by \( A^{\prime} \). Here \( f(\lambda) \) is a nonnegative integrable function in \( (-\pi,\pi] \). We shall define \( f(\lambda) \) with period \( 2\pi \) on the real axis. Some of these minimum problems arise in the theory of stationary stochastic processes. These applications will be discussed in §5 [Grenander 1951].

[13] M. Rosenblatt: “An inventory problem,” Econometrica 22 : 2 (April 1954), pp. 244–247. MR 61355 Zbl 0058.36401 article

A certain class of “inventory” problems has been discussed in recent econometric papers, in particular in the papers of Arrow, Harris, and Marschak [1951] and Dvoretzky, Kiefer, and Wolfowitz [1952]. In these papers, the authors have for the most part assumed that the organization studied has the supply of the material being stored under control in the sense that any amount of material it orders will be supplied. There are economic contexts in which the supply of material is subject to random fluctuation and is not under the control of the organization in the sense spoken of above.

This paper does not attempt to set up a general model for such problems. The object is to set up a very simple model and attempt a solution in a special case in the hope that it will point the way to solutions of problems of more practical interest. It will be convenient to speak of the model in terms of government storage of grain.

[14] U. Grenander and M. Rosenblatt: “Regression analysis of time series with stationary residuals,” Proc. Natl. Acad. Sci. U.S.A. 40 : 9 (September 1954), pp. 812–816. MR 62403 Zbl 0059.13404 article

[15] M. Rosenblatt: Some purely deterministic processes. Technical report 7, College of Engineering, New York University, 1956. A version of this was later published in J. Math. Mech. 6:4 (1957). MR 80397 techreport

[16] J. R. Blum and M. Rosenblatt: “A class of stationary processes and a central limit theorem,” Proc. Natl. Acad. Sci. U.S.A. 42 : 7 (July 1956), pp. 412–413. A longer version of this was published in Duke Math. J. 24:1 (1957). MR 81023 Zbl 0070.36403 article

[17] U. Grenander and M. Rosenblatt: “Some problems in estimating the spectrum of a time series,” pp. 77–93 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 1: Contributions to the theory of statistics. Edited by J. Nyman. University of California Press (Berkeley and Los Angeles, CA), 1956. MR 84914 Zbl 0072.36401 incollection

Jerzy Neyman	Related
Ulf Grenander	Related

[18] M. Rosenblatt: “Some regression problems in time series analysis,” pp. 165–186 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 1: Contributions to the theory of statistics. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles, CA), 1956. MR 84920 Zbl 0071.35602 incollection

Estimates of the regression coefficients which are unbiased and linear in the observations are discussed in this paper. The residual is assumed to be a stationary process. Two specific estimates are discussed, the least-squares estimate and the Markov estimate. I call the estimate which is computed under the assumption that the residual is an orthogonal process the least-squares estimate. The Markov estimate is the linear unbiased estimate with minimal covariance matrix. The basic assumptions made in the paper are discussed in section 2 and are held to throughout the paper. In section 3 some remarks about the approximation of a continuous positive definite matrix-valued function by finite trigonometric forms are made. These remarks are used in section 4 to obtain the main results about the asymptotic behavior of the covariance matrices of the least-squares and Markov estimates. The next section discusses the many interesting cases in which the least-squares estimate is asymptotically as good as the Markov estimate. The first really systematic discussion of some of these problems was given by U. Grenander [1954]. Further work was carried out by U. Grenander and M. Rosenblatt in [1954a, 1954b, 1957]. The author considers some of these problems in the case of a vector-valued time series in [1956]. Some of the results of this paper are a generalization of some of those obtained in [1956].

A few cases in which the least-squares estimate is not asymptotically efficient in the class of linear unbiased estimates are discussed in sections 5 and 7. Some small sample computations for a linear regression with a residual which is a first order autoregressive scheme are carried out in section 6 to test the asymptotic theory.

[19] M. Rosenblatt: “A central limit theorem and a strong mixing condition,” Proc. Natl. Acad. Sci. U.S.A. 42 : 1 (January 1956), pp. 43–47. MR 74711 Zbl 0070.13804 article

This paper presents a central limit theorem for a sequence of dependent random variables \[ X_1, X_2,\dots .\] The assumptions required are the usual assumptions on second and \( 2 + \delta \) order moments and a strong mixing condition. The theorem is of interest for two reasons. x‘All general central limit theorems for dependent random variables formalize in some sense a heuristic notion of A. Markoff to the effect that one expects a central limit theorem to hold for \( X_1 \), \( X_2,\dots \), if the random variables behave more like independent random variables the farther they are separated (assuming that appropriate moments exist). An interesting discussion of this intuitive notion is given in S. Bernstein’s paper on the central limit theorem [1926]. The strong mixing condition used in this paper seems to be a more intuitively appealing formalization of this notion than most others. The condition is also of interest because it is a strong version of the mixing condition encountered in ergodic theory (see [Hopf 1937, p. 35]).

[20] M. Rosenblatt: “On the estimation of regression coefficients of a vector-valued time series with a stationary residual,” Ann. Math. Stat. 27 : 1 (1956), pp. 99–121. MR 76264 Zbl 0071.13604 article

Time series which are realizations of a vector-valued stochastic process of dimension two with a stationary disturbance are considered. Linear estimates of the regression coefficients of the time series are discussed, in particular the least-squares or classical estimate and the Markov estimate. The least-squares estimate is the estimate computed under the assumption that the components of the disturbance are orthogonal processes and orthogonal to each other. It is known that the Markov estimate is in general better than the least-squares estimate. The asymptotic behavior of the covariance matrices of the least-squares estimate and of the Markov estimate is described. Conditions under which the least-squares estimate is as good asymptotically as the Markov estimate are obtained, that is, conditions under which the least-squares estimate is efficient asymptotically in the class of linear unbiased estimates. The analogues of the results described for vector-valued time series of dimension greater than two can be seen to hold.

[21] M. Rosenblatt: “Remarks on some nonparametric estimates of a density function,” Ann. Math. Stat. 27 : 3 (1956), pp. 832–837. MR 79873 Zbl 0073.14602 article

[22] U. Grenander and M. Rosenblatt: Statistical analysis of stationary time series. Wiley Publications in Mathematical Statistics. Almqvist & Wiksell (Stockholm), 1957. A 2nd, corrected edition was published in 1984, then republished in 2008. MR 84975 Zbl 0080.12904 book

[23] M. Rosenblatt: “A random model of the sea surface generated by a hurricane,” J. Math. Mech. 6 : 2 (1957), pp. 235–246. MR 85810 Zbl 0077.12801 article

The object of this paper is to construct a random model of the sea surface (fully developed) generated by a hurricane. The storm area is assumed to be fixed. Let the ocean surface at \( Re^{i\psi} \) at time \( t \) be given by \( \eta(Re^{i\psi},t) \) (a complex-valued coordinate system is used for convenience). The zero point is assumed to be the center of the storm area. It seems reasonable to require that \( \eta(Re^{i\psi},t) \) be a random process stationary in \( \psi \) and \( t \). The character of a process stationary with respect to the angular coordinate \( \psi \) is discussed in Section 2. Of course, we want the surface to satisfy the linearized equations of motion. A representation of a process \( \eta(Re^{i\psi},t) \) stationary in \( \psi \), \( f \) and satisfying the equations of motion is developed in Section 3. At a large distance from the storm center the surface should appear to be a superposition of waves diverging radially from the storm center. There must then be some singularity of the solution, the energy being fed into the sea surface at this singularity. The representation derived in Section 3 does have a singularity but does not appear to be too realistic. Two other models that appear to be more realistic are obtained in Sections 4 and 5. The character of the representations are investigated at a large distance from the storm center. One then obtains a representation that to the first order looks like Pierson’s representation of a storm-generated ocean surface [1955]. The analysis indicates that one ought to be able to obtain information about the surface by carrying out a random spectral analysis of the surface at some distance from the storm center.

[24] M. Rosenblatt: “Some purely deterministic processes,” J. Math. Mech. 6 : 4 (1957), pp. 801–810. A version of this was earlier published as a 1956 technical report. MR 93827 Zbl 0080.35001 article

The linear prediction problem as it arises in the case of stationary processes has attracted much attention. The problem has been studied most intensively when the prediction error is known to be positive. It is of some interest to consider a few simple situations in which the prediction error is zero, especially as a situation of this sort arises in Neumann’s theoretical model of storm-generated ocean waves [Pierson, Jr. 1955]. I shall, unfortunately, not be able to discuss the problem that arises in the context of Neumann’s model in any detail.

Let \( x_t \), \( Ex_t \equiv 0 \), be a weakly stationary process, that is, \[ r_{t,\tau} = Ex_tx_r = r_{t-\tau} \] depends only on the time difference \( t - \tau \). Our process is assumed to be real-valued. The time parameter \( t \) may either range over the real numbers or it may range over the integers. The first case is the continuous parameter case and the second is that of a discrete parameter. Examples of both continuous parameter and discrete parameter processes will be discussed. The discrete parameter processes discussed will be of much greater interest than the continuous parameter processes.

[25] J. R. Blum and M. Rosenblatt: “A class of stationary processes and a central limit theorem,” Duke Math. J. 24 : 1 (1957), pp. 73–78. A shorter version of this was published in Proc. Natl. Acad. Sci. U.S.A. 42:7 (1956). MR 83215 Zbl 0083.14101 article

[26] M. Rosenblatt: “The multidimensional prediction problem,” Proc. Natl. Acad. Sci. U.S.A. 43 : 11 (November 1957), pp. 989–992. A closely related paper was published in Statistical methods of radio wave propagation (1960). MR 97855 Zbl 0090.35003 article

[27] C. J. Burke and M. Rosenblatt: “A Markovian function of a Markov chain,” Ann. Math. Stat. 29 : 4 (1958), pp. 1112–1122. MR 101557 Zbl 0100.34402 article

[28] M. Rosenblatt: “A multi-dimensional prediction problem,” Ark. Mat. 3 : 5 (1958), pp. 407–424. MR 92332 Zbl 0084.35504 article

[29] J. R. Blum, H. Chernoff, M. Rosenblatt, and H. Teicher: “Central limit theorems for interchangeable processes,” Can. J. Math. 10 (1958), pp. 222–229. MR 96298 Zbl 0081.35203 article

Henry E. Teicher	Related
Herman Chernoff	Related
Julius Rubin Blum	Related

[30] M. Rosenblatt: “Functions of a Markov process that are Markovian,” J. Math. Mech. 8 : 4 (1959), pp. 585–596. MR 103539 Zbl 0100.34403 article

In this paper we are primarily concerned with discrete time parameter Markov processes \( \{X(n)\} \), \( n = 0,1 \), \( 2,\dots \), with stationary transition mechanism. The processes \[ \{Y(n)\} = \{f(X(n))\} \] generated by a given many-one function \( f \) and the processes \( \{X(n)\} \) with a fixed stationary transition mechanism are constructed. The processes \( \{Y(n)\} \) are in a one-to-one correspondence with the possible initial distributions of \( \{X(n)\} \). The object of the paper is to determine conditions under which \( \{Y(n)\} \) is Markovian, whatever the initial distribution of \( \{X(n)\} \). Necessary and sufficient conditions for the new processes \( \{Y(n)\} \) to be Markovian are obtained under the assumption that the family of measures corresponding to the fixed transition mechanism (of \( \{X(n)\} \)) is dominated [Halmos and Salvage 1949]. The conditions are expressed, of course, in terms of the function \( f(\,\cdot\,) \) and the transition mechanism. Generally the processes \( \{Y(n)\} \) do not have stationary transition mechanism. The conditions simplify in the case of a continuous time parameter Markov chain. Some of the discussions may at times have the flavor of those used in considering the concept of sufficiency [Halmos and Salvage 1949]. Some aspects of the problem discussed in the paper are touched on in [Burke and Rosenblatt 1958].

[31] M. Rosenblatt: “Statistical analysis of stochastic processes with stationary residuals,” pp. 246–275 in Probability and statistics: The Harald Cramér volume. Edited by U. Grenander. Wiley Publications in Statistics. Almqvist & Wiksell (Stockholm), 1959. MR 107954 Zbl 0201.51701 incollection

Harald Cramér	Related
Ulf Grenander	Related

[32] M. Rosenblatt: “Stationary processes as shifts of functions of independent random variables,” J. Math. Mech. 8 : 5 (1959), pp. 665–681. MR 114249 Zbl 0092.33601 article

Let \( x_n \), \( n = 0,\pm 1, \pm 2,\dots \), be a strictly stationary process. Two closely related problems are posed with respect to the structure of strictly stationary processes. In the first problem we ask whether one can construct a random variable \[ \xi_n = g(x_n,x_{n-1},\dots) ,\] a function of \( x_n \), \( x_{n-1},\dots \), that is independent of the past, that is, independent of \( x_{n-1} \), \( x_{n-2},\dots \). Such a sequence of random variables \( \{\xi_n\} \) is a sequence of independent and identically distributed random variables. Further, given such a construction, is \( x_n \) a function of \( \xi_n \), \( \xi_{n-1},\dots \). Necessary and sufficient conditions for such a representation are obtained in the case where \( x_n \) is a finite state Markov chain with the positive transition probabilities in any row of the transition probability matrix \( P = (p_{ij}) \) of \( x_n \), distinct. Such a representation is comparatively rare for a finite state Markov chain. In the second problem, the assumption that the independent and identically distributed \( \xi_n \)’s be functions of \( x_n \), \( x_{n-1},\dots \) is removed. We ask whether for some such family \( \{\xi_n\} \) there is a process \( \{y_n\} \), \( y_n = g(\xi_n \), \( \xi_{n-1},\dots) \), With the same probability structure as \( \{x_n\} \). This is shown to be the case for every ergodic finite state Markov chain with nonperiodic states. Sufficient conditions for such representations in the case of a general strictly stationary process are obtained.

[33] J. R. Blum and M. Rosenblatt: “On the structure of infinitely divisible distributions,” Pac. J. Math. 9 : 1 (1959), pp. 1–7. MR 105729 Zbl 0085.12901 article

Let \( F(x) \) be a distribution on the real line. Then we may write \[ F(x) = pF_1(x) + (1 - p)F_2(x) \] where \( F_1(x) \) is a discrete distribution, \( F_2(x) \) is a continuous distribution and \( 0 \leq p \leq 1 \). We shall say that \( F(x) \) is discrete if \( p = 1 \), \( F(x) \) is continuous if \( p = 0 \) and \( F(x) \) is a mixture if \( 0 < p < 1 \).

Let \[ \phi(s) = \int_{-\infty}^{\infty} e^{isx}dF(x) \] be the characteristic function corresponding to \( F(x) \). It would be useful to give a convenient criterion on \( \phi(s) \) to determine when the corresponding distribution \( F(x) \) is discrete, continuous, or a mixture. In §2 we give such a criterion for the class of infinitely divisible (i.d.) distributions, utilizing the Khinchin representation of the characteristic function of such a distribution. In §3 we apply the theorem of §2 to characterize a certain class of stochastic processes.

[34] C. Burke and M. Rosenblatt: “Consolidation of probability matrices,” Bull. Inst. Internat. Statist. 36 : 3 (1959), pp. 7–8. MR 120680 Zbl 0111.15005 article

[35] M. Rosenblatt: “An aggregation problem for Markov chains,” pp. 87–92 in Information and decision processes (West Lafayette, IN, April 1959). Edited by R. E. Machol. McGraw-Hill (New York), 1960. MR 116387 incollection

[36] M. Rosenblatt: “Limits of convolution sequences of measures on a compact topological semigroup,” J. Math. Mech. 9 : 2 (1960), pp. 293–305. MR 118773 Zbl 0099.34302 article

Limit properties of the convolution sequence of a regular measure on a compact topological semigroup are examined in this paper. Similar questions, as they arise in the case of a compact group, were examined by Kawada and Ito [1940]. Recently Bellman [1954] and [Grenander 1959] considered special limit theorems for products of independent identically distributed random operators. Such problems are closely related to those in this paper. It should be noted that similar questions arise when considering the structure of stationary stochastic processes [Rosenblatt 1959]. Various results on compact semigroups are used in characterizing the class of limit measures [Numakura 1952; Rosen 1956; Wallace 1956].

[37] M. Rosenblatt: “Stationary Markov chains and independent random variables,” J. Math. Mech. 9 : 6 (1960), pp. 945–949. An addendum to this article was published in J. Math. Mech. 11:2 (1962). MR 166839 Zbl 0096.34004 article

[38] M. Rosenblatt: “Asymptotic distribution of eigenvalues of block Toeplitz matrices,” Bull. Am. Math. Soc. 66 : 4 (1960), pp. 320–321. MR 124086 Zbl 0129.31205 article

Let \( g(\lambda) \), \( -\pi\leq\lambda\pi \), be a \( p{\times}p \) (\( p = 1,2,\dots \)) matrix-valued Hermitian function. Further \( g(\lambda) \) is bounded on \( [-\pi,\pi] \), that is, its elements are bounded on \( [-\pi,\pi] \). The Fourier coefficients \[ a_k = \frac{1}{2k}\int_{-\pi}^{\pi}e^{ik\lambda}g(\lambda)\,d\lambda, \quad k = 0,\pm 1,\dots, \] are then bounded in \( k \). We call the \( np{\times}np \) matrix \[ A_n = (a_{j-k}; \,j,k = 1,\dots, n) \] (an \( n{\times}n \) matrix of the \( p{\times}p \) blocks \( a_{j-k} \)) the \( n \)-th section block Toeplitz matrix generated by \( g(\lambda) \). Notice that the block Toeplitz matrix \( A_n \) is generally not Toeplitz. Our interest is in obtaining the asymptotic distribution of eigenvalues of \( A_n \) as \( n\to\infty \). The proof is suggested by an argument given in the one-dimensional case (\( p = 1 \)) [Grenander and Szegő 1958] and is based on results in the multidimensional prediction problem [Wiener and Masani 1957].

[39] M. Rosenblatt: “The multidimensional prediction problem,” pp. 99–111 in Statistical methods of radio wave propagation (Los Angeles, 18–20 June 1958). Edited by W. C. Hoffman. Pergamon Press (New York), 1960. A closely related paper was published in Proc. Natl. Acad. Sci. U.S.A. 43:11 (1957). incollection

G. Szegő discussed a minimum problem and an allied problem on the representation of a non-negative function as the absolute square of a one-sided trigonometric series in two fundamental papers [1920, 1921]. In later years it was realized that Szegő’s problem is the complete analytic couterpart of a one-dimensional statistical prediction problem (see [Doob 1953, Chapter 12], for a full discussion of the problem and references to the literature). Very recently considerable interest has arisen in matrix analogues of Szegő’s results, especially, since they are the analytic counterparts of corresponding multidimensional prediction problems. In [Helson and Lowdenslager 1957; Wiener and Masani 1957] results in this direction have been obtained. This survey paper derives similar results (with similar techniques) by reducing aspects of the multidimensional problem to the corresponding one-dimensional problem.

[40] M. Rosenblatt: “Some comments on narrow band-pass filters,” Quart. Appl. Math. 18 : 4 (1960–1961), pp. 387–393. MR 121867 Zbl 0099.34601 article

[41]M. Rosenblatt: “Independence and dependence,” pp. 431–443 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 20–30 July 1960), vol. 2. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 133863 Zbl 0105.11802 incollection

A stochastic process is commonily used as a model in studying the behavior of a random system through time. It will be convenient for us to take the stochastic process \( \{x_t\} \) as discrete in time \( t = \dots, -1 \), 0, \( 1,\dots \). Processes of independent random variables are the simplest and most completely understood. It is, however, clear that these are extremely limited in scope as models and one must have recourse to dependent processes (the random variables \( x_t \), not independent) in order to have any power in description. For simplicity, let us further restrict ourselves to processes that are stable through time, stationary processes. For such processes the probabilities of events shifted through time remain the same, that is, the probability \[ P\{x_{t_1+h} \leq a_1,\dots,x_{t_n+h} \leq a_n\} \] is independent of \( h \). Such models occur fairly often in the physical sciences. If mean properties of the process are to be capable of being estimated reasonably well from part of a realization of the process, some form of asymptotic independence for blocks of random variables of the process that are widely separated must be satisfied. This is, in effect, the gist of many of the results in ergodic theory. Two types of interesting problems are posed.

The first of these is concerned with reasonable notions of asymptotic independence and what types of processes satisfy them.

The second is that of characterizing those processes \( \{x_t\} \) that can be constructed out of independent processes by a function and its shifts, that is, \[ x_t = (\dots, \xi_{t-1},\xi_t,\xi_{t+1},\dots) \] where \( \{\xi_t\} \) is a process of independent random variables. Neither of these questions have elicited satisfactory answers. However, there are some small results that do give insights into the problems. The object of this paper is a presentation and discussion of a few of these limited results.

[42] J. R. Blum, J. Kiefer, and M. Rosenblatt: “Distribution free tests of independence based on the sample distribution function,” Ann. Math. Stat. 32 : 2 (1961), pp. 485–498. MR 125690 Zbl 0139.36301 article

Certain tests of independence based on the sample distribution function (d.f.) possess power properties superior to those of other tests of independence previously discussed in the literature. The characteristic functions of the limiting d.f.’s of a class of such test criteria are obtained, and the corresponding d.f. is tabled in the bivariate case, where the test is equivalent to one originally proposed by Hoeffding [1948]. A discussion is included of the computational problems which arise in the inversion of characteristic functions of this type. Techniques for computing the statistics and for approximating the tail probabilities are considered.

Julius Rubin Blum	Related
Jack Carl Kiefer	Related

[43] M. Rosenblatt: Random processes. University Texts in the Mathematical Sciences. Oxford University Press (New York), 1962. A 2nd edition was published in 1974. MR 133862 Zbl 0107.12204 book

[44] J. Hachigian and M. Rosenblatt: “Functions of reversible Markov processes that are Markovian,” J. Math. Mech. 11 : 6 (1962), pp. 951–960. MR 145588 Zbl 0115.13603 article

This paper extends a number of results obtained by C. J. Burke and M. Rosenblatt [1958, 1959]) on conditions under which functions of Markov processes are Markovian. In [1958] only finite state Markov chains were considered under the following conditions: either (a) the chain is stationary and reversible, or (b) the chain has a stationary transition mechanism with continuous time parameter and any initial distribution is allowed. In both these contexts, under appropriate conditions, if the transition probabilities of the collapsed chain satisfy the Chapman–Kolmogorov equation, the collapsed chain must be Markovian. It is well known that this is not generally true for Markov processes. The first counter-example was constructed by P. Lévy [1949]. In [1959] Rosenblatt characterized in some generality the conditions under which functions of Markov processes with stationary transition mechanism are Markovian, whatever the initial distribution. However, the Chapman–Kolmogorov equation and its implications were not considered.

In this paper we are primarily concerned with stationary reversible Markov processes. The results obtained are quite general. It is shown that if the transition probabilities of the collapsed process satisfy the Chapman–Kolmogorov equation, then the collapsed process must be Markovian. Examples of reversible Markov processes are given. Remarks are made about functions of a diffusion process on the circle.

[45] M. Rosenblatt: “Asymptotic behavior of eigenvalues of Toeplitz forms,” J. Math. Mech. 11 : 6 (1962), pp. 941–949. MR 150841 Zbl 0108.31205 article

[46] M. Rosenblatt: “Addendum to ‘Stationary Markov chains and independent random variables’,” J. Math. Mech. 11 : 2 (1962), pp. 317. Addendum to an article published in J. Math. Mech. 9:6 (1960). MR 166840 article

[47] W. Freiberger, M. Rosenblatt, and J. W. Van Ness: “Regression analysis of vector-valued random processes,” J. Soc. Indust. Appl. Math. 10 : 1 (March 1962), pp. 89–102. MR 137266 Zbl 0111.32902 article

John Winslow Van Ness	Related
Walter F. Freiberger	Related

[48] G. F. Newell and M. Rosenblatt: “Zero crossing probabilities for Gaussian stationary processes,” Ann. Math. Stat. 33 : 4 (1962), pp. 1306–1313. MR 141153 Zbl 0113.33401 article

Let \( X(t) \) be a separable stationary Gaussian process with mean zero, \( EX(t) \equiv 0 \), and continuous covariance function \[ \rho(t) = EX(\tau)\,X(\tau + t) \] normalized so that \( \rho(0) = 1 \). Questions relating to the probability \[ H_X(T) = P[X(t) > 0], \quad 0 \leq t\leq T\ \] that \( X(t) \) does not cross zero during some time interval \( T \) arise in various applications [Rice 1944–1945; Slepian 1962].

Many of the difficulties that arise in treating such questions are due to the fact that most of the interesting stationary Gaussian processes are not Markovian. Here we shall obtain bounds on the behavior of \( H_X(T) \) particularly for large \( T \) using an interesting inequality of D. Slepian [1962] and estimates of \( H_X(T) \) for some simple processes.

[49] M. Rosenblatt and D. Slepian: “\( N \)th order Markov chains with every \( N \) variables independent,” J. Soc. Indust. Appl. Math. 10 : 3 (September 1962), pp. 537–549. MR 150824 Zbl 0154.43103 article

[50] M. Rosenblatt: “Asymptotic behavior of eigenvalues for a class of integral equations with translation kernels,” pp. 316–326 in Proceedings of a symposium on time series analysis (Providence, RI, 11–14 June 1962). Edited by M. Rosenblatt. SIAM Series in Applied Mathematics. Wiley (New York), 1963. An extension of this paper was published in J. Math. Mech. 12:4 (1963). MR 147864 Zbl 0138.40404 incollection

[51] M. Rosenblatt: “Some results on the asymptotic behavior of eigenvalues for a class of integral equations with translation kernels,” J. Math. Mech. 12 : 4 (1963), pp. 619–628. Extension of a pape published in Proceedings of a symposium on time series analysis (1963). MR 150551 Zbl 0192.20903 article

[52] M. Rosenblatt: “The representation of a class of two state stationary processes in terms of independent random variables,” J. Math. Mech. 12 : 5 (1963), pp. 721–730. MR 156377 Zbl 0129.30104 article

[53] M. Heble and M. Rosenblatt: “Idempotent measures on a compact topological semigroup,” Proc. Am. Math. Soc. 14 : 1 (1963), pp. 177–184. MR 169971 Zbl 0199.05602 article

[54] Proceedings of a symposium on time series analysis (Providence, RI, 11–14 June 1962). Edited by M. Rosenblatt. SIAM Series in Applied Mathematics. Wiley (New York), 1963. Zbl 0119.00505 book

[55] M. Rosenblatt: “Some nonlinear problems arising in the study of random processes,” J. Res. Nat. Bur. Standards Sect. D 68D : 9 (September 1964), pp. 933–936. MR 165570 Zbl 0173.46204 article

Two problems of a nonlinear character concerned with random processes are discussed. In both cases the processes are assumed to be stationary.

The first problem is concerned with the representation of a discrete time parameter stationary random process as a one-sided function (nonlinear generally) of independent random variables and its shifts. This is a representation one might expect if the process is purely nondeterministic. Comments are made on the continuous parameter version of this problem, indicating that it is likely to be much more difficult and perhaps less important from a practical point of view. The second problem is concerned with the harmonic resolution of the moments (of degree two or higher) of stationary random processes. The harmonic resolution of third order moments (the “bispectrum”) is consid ered in some detail and remarks are made about statistical estimates of the bispectrum.

[56] M. Rosenblatt: “Almost periodic transition operators acting on the continuous functions on a compact space,” J. Math. Mech. 13 : 5 (1964), pp. 837–847. MR 169047 Zbl 0134.34804 article

Almost periodic transition probability operators \( T \) acting on continuous functions on a compact space are discussed in this paper. If \( T \) is irreducible, it is shown that the space can be partitioned into disjoint sets of points which the transformation \( T \) maps onto each other. The decomposition of the space into these “almost periodically moving sets” of points corresponds to the partition of the state space into cyclically moving sets in the case of a countable state Markov chain. If the almost periodically moving sets are mapped into points on a new space, the action of \( T \) on these sets can be identified with a continuous point transformation on the new space. In case of irreducible \( T \), the action of the operator is described on an appropriate \( L^2 \) space. It is shown that the \( L^2 \) space can be decomposed into two orthogonal subspaces; on the first \( T \) acts as an isometric operator and on the second as a dissipative operator. Generalizations of some of the results are noted in the case of an almost periodic positive operator.

[57] M. Rosenblatt: “Equicontinuous Markov operators,” Teor. Verojatnost. i Primenen. 9 : 2 (1964), pp. 205–222. Russian translation of article published in Theory Probab. Appl. 9:2 (1964). MR 171318 Zbl 0133.40101 article

[58] M. Rosenblatt: “Equicontinuous Markov operators,” Theory Probab. Appl. 9 : 2 (1964), pp. 180–197. A Russian translation of this article was published in Teor. Verojatnost. i Primenen. 9:2 (1964). article

In the paper we study limit properties of equicontinuous (nearly periodic) positive operators which transform continuous functions into continuous ones. The domain of definition of the functions is a compact Hausdorff space \( X \). Section 1 contains some preliminary information. In Section 2, positive Markov operators are considered. A decomposition of part of the space \( X \) into ergodic sub-parts is obtained, which is analogous to the decomposition of Krylov and Bogolyubov. In the next section eigenfunctions of positive operators are studied which correspond to eigenvalues with maximal absolute values. The theory of Perron–Frobenius is generalized to the situation considered. Section 4 is devoted to the investigation of the asymptotic behavior of the powers \( T^n \) of Markov transition operators. Finally, in Section 5, we consider the asymptotic behavior of the convolutions \( \nu^n \), \( n=1 \), \( 2,\dots \), of a regular measure on a compact topological subgroup. Some results obtained in the previous sections are used for the study of this question.

[59] M. Rosenblatt and J. W. Van Ness: “Estimation of the bispectrum,” Ann. Math. Stat. 36 : 4 (1965), pp. 1120–1136. MR 179902 Zbl 0135.19804 article

Recently interest has arisen in statistical applications of the bispectrum of stationary random processes. (The bispectrum can be thought of as the Fourier transform of the third-order moment function of the process.) The principal area of statistical harmonic analysis to receive attention previous to this time has been second-order (i.e. spectral) theory on which there is a vast literature. However, the spectrum is most useful in problems of a “linear nature” (see discussion beginning on p. viii of Blackman and Tukey [1959]) and provides insufficient information in nonlinear problems. A desire to study phenomena of a nonlinear character has attracted attention to the higher order theory. Such was the case, for example, in a recent study by Hasselmann, Munk and MacDonald [1963] where the bispectrum is used in connection with oceanographic problems, among which, as the authors state, a number of interesting phenomena such as surf beats, wave breaking, and the energy transfer between wave components can be explained only by the nonlinearity of the wave motion. The bispectrum therefore provides a first glimpse at the nonlinear effects. It is the purpose of the present paper to discuss estimating the weighted and unweighted bispectral density given a set of observations of the process. The relevant properties (consistency and asymptotic unbiasedness) of the estimates are derived for certain general classes of processes.

[60] M. Rosenblatt: “Products of independent identically distributed stochastic matrices,” J. Math. Anal. Appl. 11 (1965), pp. 1–10. An erratum for this article was published in J. Math. Anal. Appl. 15:2 (1966). MR 185636 Zbl 0203.19501 article

[61] M. Rosenblatt: “Remarks on higher order spectra,” pp. 383–389 in Multivariate analysis (Dayton, OH, 14–19 February 1965). Edited by P. R. Krishnaiah. Academic Press (New York), 1966. MR 211568 Zbl 0213.19601 incollection

[62] M. Rosenblatt: “Erratum: ‘Products of independent identically distributed stochastic matrices’,” J. Math. Anal. Appl. 15 : 2 (August 1966), pp. 386. Erratum for an article published in J. Math. Anal. Appl. 11 (1965). MR 198518 article

[63] M. Rosenblatt: “Functions of Markov processes,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 : 3 (1966), pp. 232–243. MR 203812 Zbl 0295.60064 article

The study of dependent phenomena in probability theory has centered about Markov processes to a great extent, partly because of the relatively simple structure of these processes. In many cases, particularly in applications in physics [Kac 1959] and economics [Rosenblatt 1965], one considers a function of the Markov processes and it seems to be rather natural to inquire under what circumstances (in terms of the function and the original process) the derived process will retain some aspect of the Markov property. Specifically we wish to consider the following two problems: 1. When will the first order transition probabilities of the derived process still satisfy the Chapman–Kolmogorov equation. 2. When will the derived process be Markovian itself. Further, one is naturally led to the question of the relationship between the Chapman–Kolmogorov equation and the full Markov property. The first order transition probabilities of a Markov process always satisfy the Chapman-Kolmogorov equation. However, there are many examples of non-Markovian processes whose first order transition probabilities do satisfy the Chapman–Kolmogorov equation [Feller 1959; Hachigian 1963; Lévy 1949; Rosenblatt and Slepian 1962]. Some of these problems appear to be difficult. There is a small literature on these problems [Burke and Rosenblatt 1958; Dynikin 1965; Feller 1959; Hachigian 1963; Hachigian and Rosenblatt 1962; Rosenblatt 1965, 1959; Rosenblatt and Slepian 1962; Sarmanov and Zakharov 1963]. We shall derive some of the results already in the literature together with some interesting new results in a manner that we hope may cast further light on these problems.

[64] M. Rosenblatt: “Remarks on ergodicity of stationary irreducible transient Markov chains,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6 : 4 (December 1966), pp. 293–301. MR 210192 Zbl 0146.38405 article

Consider a stationary process \[ \{X_n(\omega),\,-\infty < n < \infty\} .\] If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process \[ \{X_n(\omega),\,-\infty < n < \infty\} \] and of each of the subprocesses \[ \{X_n(\omega),\,0 \leq n < \infty\} \quad\text{and}\quad \{X_n(\omega),\,-\infty < n \leq 0\} \] are equivalent (see [Doob 1953]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain \[ \{X_n(\omega),\,-\infty < n < \infty\} \] with \[ \{X_n(\omega),\,0 \leq n < \infty\} \quad\text{ergodic} \] but \[ \{X_n(\omega),\,-\infty < n \leq 0\} \quad\text{nonergodic} \] is given. That this can be the case has already been implicitly indicated in the literature [Hunt 1960]. Another example of a stationary irreducible transient Markov chain with both \[ \{X_n(\omega),\,0 \leq n < \infty\} \quad\text{and}\quad \{X_n(\omega),\,-\infty < n \leq 0\} \] ergodic but \[ \{X_n(\omega),\, -\infty < n < \infty\} \] nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains \[ \{X_n(\omega),\,-\infty < n < \infty\} \] are nonergodic.

[65] D. R. Brillinger and M. Rosenblatt: “Asymptotic theory of estimates of \( k \)-th order spectra,” pp. 153–188 in Advanced seminar on spectral analysis of time series (Madison, WI, 3–5 October 1966). Edited by B. Harris. John Wiley (New York), 1967. A shorter version of this was published in Proc. Natl. Acad. Sci. U.S.A. 57:2 (1967). MR 211566 Zbl 0157.47402 incollection

David Brillinger	Related
Bernard Harris	Related

[66] D. R. Brillinger and M. Rosenblatt: “Computation and interpretation of \( k \)-th order spectra,” pp. 189–232 in Advanced seminar on spectral analysis of time series (Madison, WI, 3–5 October 1966). Edited by B. Harris. John Wiley (New York), 1967. MR 211567 Zbl 0157.47403 incollection

David Brillinger	Related
Bernard Harris	Related

[67] M. Rosenblatt: “A strong mixing condition and a central limit theorem on compact groups,” J. Math. Mech. 17 : 2 (1967), pp. 189–198. A corrigendum to this article was published in J. Math. Mech. 17:9 (1968). MR 212845 Zbl 0153.47304 article

A particular type of “strong mixing” implies the validity of the central limit theorem for dependent processes under appropriate further restraints [Ibragimov 1962; Rosenblatt 1956, 1961]. Necessary conditions and sufficient conditions for this type of strong mixing in the case of Gaussian stationary processes have been derived [Ibragimov 1965; Kolmogorov and Rozanov 1960]. This paper obtains necessary and sufficient conditions for strong mixing and a related concept of uniform ergodicity in the case of a stationary process on a compact commutative group generated by taking the product of independent identically distributed elements of the group. Sufficient conditions for strong mixing are obtained when the group is not commutative. The conditions are then used to obtain a central limit theorem on the group.

[68] M. Rosenblatt: “Transition probability operators,” pp. 473–483 in Proceedings of the fifth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1965 and 27 December 1965–7 January 1966), vol. 2: Contributions to probability theory, part 2. Edited by L. M. Le Cam and J. Neyman. University of California Press (Berkeley, CA), 1967. MR 212877 Zbl 0222.60042 incollection

Lucien Marie Le Cam	Related
Jerzy Neyman	Related

[69] D. R. Brillinger and M. Rosenblatt: “Asymptotic theory of estimates of \( k \)th-order spectra,” Proc. Natl. Acad. Sci. U.S.A. 57 : 2 (February 1967), pp. 206–210. A longer version of this was published in Advanced seminar on spectral analysis of time series (1967). MR 207021 Zbl 0146.40805 article

[70] M. Rosenblatt: “Corrigendum on: ‘A strong mixing condition and a central limit theorem on group’,” J. Math. Mech. 17 : 9 (1968), pp. 919. Corrigendum to an article published in J. Math. Mech. 17:2 (1967). MR 219101 article

[71] M. Rosenblatt: “Remarks on the Burgers equation,” J. Math. Phys. 9 : 7 (1968), pp. 1129–1136. MR 264252 Zbl 0159.15003 article

Periodic and aperiodic solutions of the Burgers equation \[ u_t + uu_x = \mu u_{xx} ,\] \( \mu > 0 \), are studied in this paper. A harmonic analysis of the solutions is carried out and the form of the spectrum is estimated for large time. Corresponding estimates of energy decay are also made. In Burgers’ work on this equation, the case in which \( \mu \downarrow 0 \) with \( t \) fixed, and one then lets \( t \to \infty \), is studied. In our investigation, a fixed value of \( \mu > 0 \) is taken and then one lets \( t \to \infty \). A similar analysis is also carried out for an irrotational solution of a similar 3-dimensional system of equations. For large time and moderate wavenumbers there is, to the first order, a drift of spectral mass from low wavenumbers to higher wavenumbers. Comments are also made on the asymptotic distribution of a class of random solutions.

[72] M. Rosenblatt: “Conditional probability density and regression estimators,” pp. 25–31 in Multivariate analysis, II (Dayton, OH, 17–22 June 1968). Edited by P. R. Krishnaiah. Academic Press (New York), 1969. MR 254987 incollection

[73] M. Rosenblatt: “Stationary measures for random walks on semigroups,” pp. 209–220 in Semigroups (Detroit, MI, 27–29 June 1968). Edited by K. W. Folley. Academic Press (New York), 1969. MR 260037 Zbl 0188.49105 incollection

[74] M. Rosenblatt: “A prediction problem and central limit theorems for stationary Markov sequences,” pp. 99–114 in Proceedings of the twelfth biennial seminar of the Canadian Mathematics Congess on time series and stochastic processes: Convexity and combinatorics (Vancouver, BC, 11–27 August 1969). Edited by R. Pyke. Canadian Mathematical Congress (Montreal), 1970. MR 273678 Zbl 0296.60025 incollection

[75] M. Rosenblatt: “Density estimates and Markov sequences,” pp. 199–213 in Nonparametric techniques in statistical inference (Bloomington, IN, 1–6 June 1969). Edited by M. L. Puri. Cambridge University Press (London), 1970. MR 278468 incollection

Estimates of the density function of a population based on a sample of independent observations have been considered in a number of papers [Bartlett 1963; Parzen 1962; Rosenblatt 1956]. Questions of bias, variance and asymptotic distribution of the estimates have been dealt with at greatest length. Our object is to look at such estimates of the density function when the observations are dependent. The results will not be dealt with in the most general context or under very general conditions. To obtain results in a simple and readily understandable form, the observations are assumed to be sampled from a stationary Markov sequence with a fairly strong condition on the Markov transition operator. However, the extent to which some of the conditions can be obviously relaxed will be indicated.

It should be noted that the asymptotic results we obtain in the case of dependent observations are essentially the same as those in the case of independent observations. This initially is surprising because it is certainly not true when estimating the distribution function by means of the sample distribution function. The more complicated nature of asymptotic results for this problem in the case of dependence can be seen in [Billingsley 1968]. However, the happy fact that the results we obtain in estimating the density have the same character as in the case of independence is due to the local character of the estimates.

[76] M. Rosenblatt: Markov processes: Structure and asymptotic behavior. Die Grundlehren der mathematischen Wissenschaften 184. Springer (Berlin), 1971. MR 329037 Zbl 0236.60002 book

[77] M. Rosenblatt: “Curve estimates,” Ann. Math. Stat. 42 : 6 (1971), pp. 1815–1842. MR 301851 Zbl 0231.62100 article

There is a large class of problems in which the estimation of curves arises naturally (see [Frenkiel and Klebanoff 1965; Van Atta and Chen 1968]). It is curious that one of the earliest extensive investigations of this type involves the estimation of the spectral density function when sampling from a stationary sequence [Bartlett 1950; Grenander and Rosenblatt 1957; Rosenblatt 1959; Szegő 1959]. Even though the simple histogram has been used for years, it was only later that the simpler question of estimating a probablility density function was dealt with at some length [Rosenblatt 1956; Parzen 1962; Čencov 1962]. Because the final character of the usual results obtained in both problem areas is quite similar, and the arguments are much more transparent in the case of the probability density function, we shall develop the results for the probability density function first. Later some corresponding results for spectra will be given. The similarities and differences in the two areas will be noted. Since the literature is rather extensive by now, any presentation of theory as given can only be a selection of topics and cannot claim to be exhaustive or perhaps even representative. There are a number of attractive open problems that one can suggest solutions to on heuristic grounds. A few of these problems will be examined. In most cases it is clear that one will not use the techniques to be proposed in estimating a density function unless there is a good deal of data (many obsevations), little a priori information about the density function, even if it is fairly crude.

[78] M. Rosenblatt: “Central limit theorem for stationary processes,” pp. 551–561 in Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1970), vol. 2: Probability theory. Edited by L. M. Le Cam, J. Neyman, and E. L. Scott. University of California (Berkeley), 1972. MR 402869 Zbl 0255.60026 incollection

Elizabeth Leonard Scott	Related
Lucien Marie Le Cam	Related
Jerzy Neyman	Related

[79] Statistical models and turbulence (La Jolla, CA, 15–21 July 1971). Edited by M. Rosenblatt and C. Van Atta. Lecture Notes in Physics 12. Springer (Berlin), 1972. MR 438885 Zbl 0227.76079 book

[80] M. Rosenblatt: “Probability limit theorems and some questions in fluid mechanics,” pp. 27–40 in Statistical models and turbulence (La Jolla, CA, 15–21 July 1971). Edited by M. Rosenblatt and C. Van Atta. Lecture Notes in Physics 12. Springer (Berlin), 1972. A report on this conference was published in Int. Stat. Rev. 40:2 (1972). MR 445584 Zbl 0229.76041 incollection

[81] M. Rosenblatt: “Uniform ergodicity and strong mixing,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 24 : 1 (1972), pp. 79–84. MR 322941 Zbl 0231.60050 article

[82] M. Rosenblatt and C. W. Van Atta: “‘Statistical models and turbulence’: Comments on turbulence and a report on the satellite symposium,” Int. Stat. Rev. 40 : 2 (August 1972), pp. 209–214. The relevant conference proceedings were also published in 1972. Zbl 0244.76028 article

[83] P. Bickel and M. Rosenblatt: “Two-dimensional random fields,” pp. 3–15 in Multivariate analysis, III (Dayton, OH, 19–24 June 1972). Edited by P. R. Krishnaiah. Academic Press (New York), 1973. MR 348832 Zbl 0297.60020 incollection

Peter J. Bickel	Related
Paruchuri R. Krishnaiah	Related

[84] P. J. Bickel and M. Rosenblatt: “On some global measures of the deviations of density function estimates,” Ann. Stat. 1 : 6 (1973), pp. 1071–1095. Corrections to this article were published in Ann. Stat. 3:6 (1975). MR 348906 Zbl 0275.62033 article

We consider density estimates of the usual type generated by a weight function. Limt theorems are obtained for the maximum of the normalized deviation of the estimate from its expected value, and for quadratic norms of the same quantity. Using these results we study the behavior of tests of goodness-of-fit and confidence regions based on these statistics. In particular, we obtain a procedure which uniformly improves the chi-square goodness-of-fit test when the number of observations and cells is large and yet remains insensitive to the estimation of nuisance parameters. A new limit theorem for the maximum absolute value of a type of nonstationary Gaussian process is also proved.

[85] M. Rosenblatt: “Invariant and subinvariant measures of transition probability functions acting on continuous functions,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 : 3 (September 1973), pp. 209–221. A correction to this article was published in Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28:1 (1973–1974). MR 339336 Zbl 0255.60052 article

[86] M. Rosenblatt: “Correction to: ‘Invariant and subinvariant measures of transition probability functions acting on continuous functions’,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 28 : 1 (December 1973), pp. 84. Correction to an article published in Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25:3 (1972–1973). MR 356236 article

[87] M. Rosenblatt: Random processes, 2nd edition. Graduate Texts in Mathematics 17. Springer (New York), 1974. 2nd edition of 1962 original. MR 346883 Zbl 0287.60031 book

[88] M. Rosenblatt: “Recurrent points and transition functions acting on continuous functions,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 30 : 3 (September 1974), pp. 173–183. MR 358996 Zbl 0279.60067 article

[89] K. S. Lii, M. Rosenblatt, and C. Van Atta: “Bispectral measurements in turbulence, I,” J. Fluid Mech. 77 : 1 (September 1975), pp. 45–62. article

Charles William Van Atta	Related
Keh-Shin Lii	Related

[90] K.-S. Lii and M. Rosenblatt: “Asymptotic results on a spline estimate of a probability density,” pp. 77–85 in Statistical inference and related topics (Bloomington, IN, 31 July–9 August 1974), vol. 2. Edited by M. L. Puri. Academic Press (New York), 1975. Volume dedicated to Z. W. Birnbaum. MR 373139 Zbl 0341.62033 incollection

Zygmunt William Birnbaum	Related
Madan Lal Puri	Related
Keh-Shin Lii	Related

[91] M. Rosenblatt: “Multiply schemes and shuffling,” Math. Comput. 29 : 131 (June 1975), pp. 929–934. MR 381231 Zbl 0322.65004 article

[92] P. J. Bickel and M. Rosenblatt: “Corrections to: ‘On some global measures of the deviations of density function estimates’,” Ann. Stat. 3 : 6 (1975), pp. 1370. Corrections to an article published in Ann. Stat. 1:6 (1973). MR 383633 Zbl 0318.62028 article

[93] M. Rosenblatt: “A quadratic measure of deviation of two-dimensional density estimates and a test of independence,” Ann. Stat. 3 : 1 (1975), pp. 1–14. Corrections to this article were published in Ann. Stat. 10:2 (1975). MR 428579 Zbl 0325.62030 article

[94] K. S. Lii and M. Rosenblatt: “Asymptotic behavior of a spline estimate of a density function,” Comput. Math. Appl. 1 : 2 (June 1975), pp. 223–235. MR 386134 Zbl 0364.62037 article

[95] M. Rosenblatt: “The local behavior of the derivative of a cubic spline interpolator,” J. Approx. Theory 15 : 4 (December 1975), pp. 382–387. MR 393952 Zbl 0321.65004 article

[96] M. Rosenblatt: “Note correcting a remark in a paper of Karl Bosch,” Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 33 : 3 (September 1975), pp. 219. MR 405587 Zbl 0314.60051 article

[97] M. Rosenblatt: “Asymptotics and representation of cubic splines,” J. Approximation Theory 17 : 4 (August 1976), pp. 332–343. An erratum to this article was published in J. Approx. Theory 28:2 (1980). MR 417632 Zbl 0351.41005 article

The local asymptotic behavior of a cubic spline interpolator (with equal bin size) and its derivatives is determined to the first order precisely in the interior as the bin size tends to zero. It is shown that this asymptotic behavior is independent of the boundary conditions usually made use of (in the case of spline interpolation on a finite interval). However, if the local behavior at the boundary or global asymptotic behavior is of interest, the type of boundary conditions assumed can determine what happens. Precise estimates of global asymptotic behavior are also derived. An explicit and simple representation is given for the cubic spline interpolator on the infinite line as well as the doubly cubic spline interpolator on the infinite plane.

[98] M. Rosenblatt: “Fractional integrals of stationary processes and the central limit theorem,” J. Appl. Probab. 13 : 4 (December 1976), pp. 723–732. MR 423496 Zbl 0354.60010 article

[99] M. Rosenblatt: “On the maximal deviation of \( k \)-dimensional density estimates,” Ann. Probab. 4 : 6 (1976), pp. 1009–1015. MR 428580 Zbl 0369.62028 article

[100] J. Rice and M. Rosenblatt: “Estimation of the log survivor function and hazard function,” Sankhyā Ser. A 38 : 1 (January 1976), pp. 60–78. MR 468018 Zbl 0398.62081 article

[101] K. N. Helland, K. S. Lii, and M. Rosenblatt: “Bispectra of atmospheric and wind tunnel turbulence,” pp. 223–248 in Application of Statistics (Dayton, OH, 14–20 June 1976). Edited by P. R. Krishnaiah. North-Holland (Amsterdam), 1977. incollection

K. N. Helland	Related
Keh-Shin Lii	Related
Paruchuri R. Krishnaiah	Related

[102] P. A. W. Lewis, L. H. Liu, D. W. Robinson, and M. Rosenblatt: “Empirical sampling study of a goodness of fit statistic for density function estimation,” pp. 159–174 in Multivariate analysis IV (Dayton, OH, 16–21 June 1975). Edited by P. R. Krishnaiah. 1977. Zbl 0399.62039 incollection

The distribution of a measure of the distance between a probability density function and its estimate is examined through empirical sampling methods. The estimate of the density function is that proposed by Rosenblatt using sums of weight functions centered at the observed values of the random variables. The weight function in all cases was triangular, but both uniform and Cauchy densities were tried for different sample sizes and bandwidths. The simulated distributions look as if they could be approximated by Gamma distributions, in many cases. Some assessment can also be made of the rate of convergence of the moments and the distribution of the measure to the limiting moments and distribution, respectively.

Paruchuri R. Krishnaiah	Related
Peter A. W. Lewis	Related
Li Hung Liu	Related
Derek W. Robinson	Related

[103] Studies in probability theory. Edited by M. Rosenblatt. MAA Studies in Mathematics 18. Mathematical Association of America (Washington, DC), 1978. MR 534847 book

[104] M. Rosenblatt: “Energy transfer for the Burgers’ equation,” Phys. Fluids 21 : 10 (1978), pp. 1694–1697. MR 521801 Zbl 0396.76021 article

[105] M. Rosenblatt: “Dependence and asymptotic independence for random processes,” pp. 24–45 in Studies in probability theory. Edited by M. Rosenblatt. MAA Studies in Mathematics 18. Mathematical Association of America (Washington, DC), 1978. MR 534849 Zbl 0406.60048 incollection

[106] K. N. Helland and M. Rosenblatt: “Spectral variance estimation and the analysis of turbulence,” Phys. Fluids 22 : 5 (May 1979), pp. 819–823. article

The results of an asymptotic analysis for the variance of power spectra and the real and imaginary parts of complex cross spectra are presented. The formulae for the spectral variances do not require an assumption of jointly Gaussian probability density functions for the time series and, therefore, are of special importance to the analysis of turbulence data. The variances computed from the asymptotic formulae compare favorably with directly estimated variances for power spectra and a third-order cross spectrum which appear in the study of spectral energy transfer in turbulent flows. The asymptotic variance formulae can be useful tools for the design of experiments which require estimation of power and cross spectra.

[107] Smoothing techniques for curve estimation (Heidelberg, 2–4 April 1979). Edited by T. Gasser and M. Rosenblatt. Lecture Notes in Mathematics 757. Springer (Berlin), 1979. MR 564249 Zbl 0407.00010 book

[108] M. Rosenblatt: “Some remarks on a mixing condition,” Ann. Probab. 7 : 1 (1979), pp. 170–172. A correction to this article was published in Ann. Probab. 7:6 (1979). MR 515825 Zbl 0925.60052 article

[109] Y. P. Mack and M. Rosenblatt: “Multivariate \( k \)-nearest neighbor density estimates,” J. Multivar. Anal. 9 : 1 (March 1979), pp. 1–15. MR 530638 Zbl 0406.62023 article

[110] M. Rosenblatt: “Some limit theorems for partial sums of quadratic forms in stationary Gaussian variables,” Z. Wahrsch. Verw. Gebiete 49 : 2 (1979), pp. 125–132. MR 543988 Zbl 0388.60048 article

[111] M. Rosenblatt: “Correction to: ‘Some remarks on a mixing condition’,” Ann. Probab. 7 : 6 (1979), pp. 1097. Correction to an article published in Ann. Probab. 7:1 (1979). MR 548908 article

[112] K. N. Helland, K. S. Lii, and M. Rosenblatt: “Bispectra and energy transfer in grid-generated turbulence,” Chapter 3, pp. 123–155 in Developments in statistics, vol. 2. Edited by P. R. Krishnaiah. Developments in statistics. Academic Press (New York), 1979. MR 554179 Zbl 0478.76060 incollection

K. N. Helland	Related
Keh-Shin Lii	Related
Paruchuri R. Krishnaiah	Related

[113] M. Rosenblatt: “Global measures of deviation for kernel and nearest neighbor density estimates,” pp. 181–190 in Smoothing techniques for curve estimation (Heidelberg, 2–4 April 1979). Edited by T. Gasser and M. Rosenblatt. Lecture Notes in Mathematics 757. Springer (Berlin), 1979. MR 564258 Zbl 0417.62030 incollection

A number of estimates of the probability density function (and regression function) have been introduced in the past few decades. The oldest are the kernel estimates and more recently nearest neighbor estimates have attracted attention. Most investigations have dealt with the local behavior of the estimates. There has, however, been some research and some heuristic comment on the utility of global measures of deviation like mean square deviation. Here, it is suggested that in a certain setting such global measures of deviation for kernel estimates may depend far less on tail behavior of the density function than in the case of nearest neighbor estimates. This appears to be due to the unstable behavior of the bias of nearest neighbor density estimates in the tails.

[114] M. Rosenblatt: “Linearity and nonlinearity in time series: Prediction,” pp. 423–434 in Proceedings of the 42nd session of the International Statistical Institute (Manila, Philippines, 4–14 December 1979), published as Bull. Inst. Internat. Statist. 48 : 1. International Statistical Institute (The Hague), 1979. MR 731551 incollection

[115] M. Rosenblatt: “Linear processes and bispectra,” J. Appl. Probab. 17 : 1 (March 1980), pp. 265–270. MR 557456 Zbl 0423.60043 article

A linear process is generated by applying a linear filter to independent, identically distributed random variables. Only the modulus of the frequency response function can be estimated if only the linear process is observed and if the independent identically distributed random variables are Gaussian. In this case a number of distinct but related problems coalesce and the usual discussion of these problems assumes, for example, in the case of a moving average that the zeros of the polynomial given by the filter have modulus greater than one. However, if the independent identically distributed random variables are non-Gaussian, these problems become distinct and one can estimate the transfer function under appropriate conditions except for a possible linear phase shift by using higher-order spectral estimates.

[116] M. Rosenblatt: “Some limit theorems for partial sums of stationary sequences,” pp. 239–248 in Multivariate analysis, V (Amsterdam). Edited by P. R. Krishnaiah. North-Holland, 1980. MR 566342 Zbl 0422.60018 incollection

[117] M. Rosenblatt: “Erratum: ‘Asymptotics and representation of cubic splines’,” J. Approx. Theory 28 : 2 (1980), pp. 184. Erratum to an article published in J. Approx. Theory 17:4 (1976). MR 573331 Zbl 0425.41011 article

[118] M. Rosenblatt: “Limit theorems for Fourier transforms of functionals of Gaussian sequences,” Z. Wahrsch. Verw. Gebiete 55 : 2 (1981), pp. 123–132. MR 608012 Zbl 0447.60016 article

[119] M. Rosenblatt: “Polynomials in Gaussian variables and infinite divisibility?,” pp. 139–142 in Contributions to probability: A collection of papers dedicated to Eugene Lukacs. Edited by J. Gani and V. K. Rohatgi. Academic Press (New York), 1981. MR 618683 Zbl 0533.60033 incollection

Vijay Kumar Rohatgi	Related
Eugene Lukacs	Related
Joseph Mark Gani	Related

[120] J. Rice and M. Rosenblatt: “Integrated mean squared error of a smoothing spline,” J. Approx. Theory 33 : 4 (December 1981), pp. 353–369. MR 646156 Zbl 0516.41006 article

[121] M. Rosenblatt: “Corrections: ‘A quadratic measure of deviation of two-dimensional density estimates and a test of independence’,” Ann. Stat. 10 : 2 (1982), pp. 646. Corrections to an article published in Ann. Stat. 3:1 (1975). MR 653543 Zbl 0503.62034 article

[122] J. Rice and M. Rosenblatt: “Boundary effects on the behavior of smoothing splines,” pp. 635–643 in Statistics and probability: Essays in honor of C. R. Rao. Edited by G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh. North-Holland (Amsterdam), 1982. MR 659512 Zbl 0524.41008 incollection

Gopinath Kallianpur	Related
Calyampudi Radhakrishna Rao	Related
Paruchuri R. Krishnaiah	Related
Jayanta Kumar Ghosh	Related
John Rischard Rice	Related

[123] K. S. Lii, K. N. Helland, and M. Rosenblatt: “Estimating three-dimensional energy transfer in isotropic turbulence,” J. Time Ser. Anal. 3 : 1 (1982), pp. 1–28. MR 660393 Zbl 0501.76044 article

Keh-Shin Lii	Related
K. N. Helland	Related

[124] K. S. Lii and M. Rosenblatt: “Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes,” Ann. Stat. 10 : 4 (1982), pp. 1195–1208. MR 673654 Zbl 0512.62090 article

[125] J. Rice and M. Rosenblatt: “Smoothing splines: Regression, derivatives and deconvolution,” Ann. Stat. 11 : 1 (1983), pp. 141–156. MR 684872 Zbl 0535.41019 article

[126] M. Rosenblatt: “Linear random fields,” pp. 299–309 in Studies in econometrics, time series, and multivariate statistics. Edited by S. Karlin, T. Amemiya, and L. A. Goodman. Academic Press (New York), 1983. Volume in commemoration of T. W. Anderson’s 65th birthday. MR 738658 Zbl 0594.60052 incollection

Takeshi Amemiya	Related
Theodore W. Anderson	Related
Leo A. Goodman	Related
Samuel Karlin	Related

[127] M. Rosenblatt: “Cumulants and cumulant spectra,” pp. 369–382 in Time series in the frequency domain. Edited by D. R. Brillinger and P. R. Krishnaiah. Handbook of Statistics 3. North-Holland (Amsterdam), 1983. MR 749794 Zbl 0539.93071 incollection

David Brillinger	Related
Paruchuri R. Krishnaiah	Related

[128] M. Rosenblatt: “Asymptotic normality, strong mixing and spectral density estimates,” Ann. Probab. 12 : 4 (1984), pp. 1167–1180. MR 757774 Zbl 0545.62058 article

[129] K. S. Lii and M. Rosenblatt: “Remarks on non-Gaussian linear processes with additive Gaussian noise,” pp. 185–197 in Robust and nonlinear time series analysis (Heidelberg, September 1983). Lecture Notes in Statistics 26. Springer (New York), 1984. MR 786308 Zbl 0568.62078 incollection

[130] K. S. Lii and M. Rosenblatt: “Non-Gaussian linear processes, phase and deconvolution,” pp. 51–58 in Statistical signal processing (Annapolis, MD, 11–15 May 1982). Edited by E. J. Wegman and J. G. Smith. Statistics: Textbooks and Monographs 53. Marcel Dekker (New York), 1984. MR 787247 Zbl 0563.62067 incollection

James G. Smith	Related
Edward Joseph Wegman	Related
Keh-Shin Lii	Related

[131] M. Rosenblatt: “A two-dimensional smoothing spline and a regression problem,” pp. 915–931 in Limit theorems in probability and statistics (Veszprém, Hungary, 21–26 June 1982), vol. 2. Edited by P. Révész. Colloquia Mathematica Societatis János Bolyai 36. North-Holland (Amsterdam), 1984. MR 807590 Zbl 0584.62104 incollection

[132] U. Grenander and M. Rosenblatt: Statistical analysis of stationary time series, 2nd, corrected edition. Chelsea Publishing Company (New York), 1984. 2nd, corrected edition of 1957 original. Republished in 2008. MR 890514 Zbl 0575.62080 book

[133] M. Rosenblatt: “Short-range and long-range dependence,” pp. 509–520 in Statistics: An appraisal (Ames, IA, 13–15 June 1983). Edited by H. T. David and H. A. David. Iowa State University Press (Ames, IA), 1984. incollection

Herbert Theodore David	Related
Herbert Aron David	Related

[134] Errett Bishop: Reflections on him and his research (San Diego, CA, 24 September 1983). Edited by M. Rosenblatt. Contemporary Mathematics 39. American Mathematical Society (Providence, RI), 1985. MR 788162 Zbl 0579.01015 book

[135] M. Rosenblatt and F. J. Samaniego: “Julius R. Blum, 1922–1982,” Ann. Stat. 13 : 1 (1985), pp. 1–9. MR 773151 Zbl 0561.01016 article

Julius Rubin Blum	Related
Francisco J. Samaniego	Related

[136] K. S. Lii and M. Rosenblatt: “A fourth-order deconvolution technique for non-Gaussian linear processes,” pp. 395–410 in Multivariate analysis VI (Pittsburgh, PA, 25–29 July 1983). Edited by P. R. Krishnaiah. North-Holland (Amsterdam), 1985. MR 822309 Zbl 0587.60035 incollection

Keh-Shin Lii	Related
Paruchuri R. Krishnaiah	Related

[137] M. Rosenblatt: Stationary sequences and random fields. Birkhäuser (Boston), 1985. MR 885090 Zbl 0597.62095 book

[138] M. Rosenblatt: “Parameter estimation for finite-parameter stationary random fields,” pp. 311–318 in Essays in time series and allied processes: Papers in honour of E. J. Hannan, published as J. Appl. Probab. 23A. Issue edited by J. Gani and M. B. Priestley. Applied Probability Trust (Sheffield, UK), 1986. MR 803180 Zbl 0606.62105 incollection

Maurice Bertram Priestley	Related
Edward James Hannan	Related
Joseph Mark Gani	Related

[139] K.-S. Lii and M. Rosenblatt: “Deconvolution of non-Gaussian linear processes with vanishing spectral values,” Proc. Natl. Acad. Sci. U.S.A. 83 : 2 (February 1986), pp. 199–200. MR 822711 Zbl 0582.60052 article

[140] M. Rosenblatt: “Convolution sequences of measures on the semigroup of stochastic matrices,” pp. 215–220 in Random matrices and their applications (Brunswick, ME, 17–23 July 1984). Edited by J. E. Cohen, H. Kesten, and C. M. Newman. Contemporary Mathematics 50. American Mathematical Society (Providence, RI), 1986. MR 841094 Zbl 0589.15014 incollection

Harry Kesten	Related
Joel E. Cohen	Related
Charles Michael Newman	Related

[141] M. Rosenblatt: “Prediction for some non-Gaussian autoregressive schemes,” Adv. Appl. Math. 7 : 2 (June 1986), pp. 182–198. An erratum to this article was published in Adv. Appl. Math. 10:1 (1989). MR 845375 Zbl 0604.62092 article

[142] K. S. Lii and M. Rosenblatt: “Estimation of a transfer function in a non-Gaussian context,” pp. 49–51 in Function estimates (Arcata, CA, 28 July–3 August 1985). Edited by J. S. Marron. Contemporary Mathematics 59. American Mathematical Society (Providence, RI), 1986. MR 870447 Zbl 0646.62077 incollection

Keh-Shin Lii	Related
James Stephen Marron	Related

[143] M. Rosenblatt: “Non-Gaussian sequences and deconvolution,” pp. 349–353 in Proceedings of the 1st world congress of the Bernoulli Society (Tashkent, USSR, 8–14 September 1986), vol. 2. Edited by Yu. A. Prohorov and V. V. Sazonov. VNU Science Press (Utrecht, The Netherlands), 1987. MR 1092477 Zbl 0671.62094 incollection

Yu. A. Prokhorov	Related
Vyacheslav Vasilevich Sazonov	Related

[144] M. Rosenblatt: “Remarks on limit theorems for nonlinear functionals of Gaussian sequences,” Probab. Theory Related Fields 75 : 1 (1987), pp. 1–10. MR 879549 Zbl 0589.60028 article

[145] M. Rosenblatt: “Some models exhibiting non-Gaussian intermittency,” IEEE Trans. Inform. Theory 33 : 2 (1987), pp. 258–262. MR 880167 article

[146] M. Rosenblatt: “Scale renormalization and random solutions of the Burgers equation,” J. Appl. Probab. 24 : 2 (June 1987), pp. 328–338. MR 889797 Zbl 0624.60071 article

[147] K.-S. Lii and M. Rosenblatt: “Estimation and deconvolution when the transfer function has zeros,” J. Theoret. Probab. 1 : 1 (1988), pp. 93–113. MR 916486 Zbl 0668.62071 article

The problem of estimation of the transfer function and deconvolution of a linear process is considered. This paper specifically deals with the case when the transfer function has zeros on the unit circle or equivalently the spectral density function has zeros. It is shown that if the zeros are finitely many and are of finite order then we can still consistently estimate the transfer function without the minimum phase assumption when the process is non-Gaussian. Statistical properties of the estimate are given. Convergence of the deconvolution is also given. It is shown that if the transfer function vanishes on an interval, then, essentially, we cannot identify the transfer function. Two simple simulated examples are given to illustrate the procedures.

[148] J. A. Rice and M. Rosenblatt: “On frequency estimation,” Biometrika 75 : 3 (September 1988), pp. 477–484. MR 967586 Zbl 0654.62077 article

[149] K.-S. Lii and M. Rosenblatt: “Nonminimum phase non-Gaussian deconvolution,” J. Multivar. Anal. 27 : 2 (November 1988), pp. 359–374. Republished in a 1989 print book version of this volume. MR 970960 Zbl 0658.60069 article

A procedure for deconvolution of nonminimum phase non-Guassian time series based on the estimation of higher order (greater than two) spectra is given. This can be applied to the analysis of seismograms. The procedure allows estimation of the wavelet. Knowledge of cumulant spectra of order greater than two allows estimation of the phase of the wavelet. In this way one has access to information not available in the ordinary second-order deconvolution procedures. Computational details of the method for estimating the phase of the wavelet are given. There are simulated illustrative examples. One of the examples is based on an actual reflectivity series from a sonic well log. The method is effective asymptotically in the nonminimum phase non-Gaussian context where the Wiener–Levinson procedure does not apply.

[150] M. Rosenblatt: “A note on maximum entropy,” pp. 255–260 in Probability, statistics, and mathematics: Papers in honor of Samuel Karlin. Edited by T. W. Anderson, K. B. Athreya, and D. L. Iglehart. Academic Press (Boston), 1989. MR 1031290 Zbl 0682.60030 incollection

Theodore W. Anderson	Related
Donald L. Iglehart	Related
Samuel Karlin	Related
Krishna B. Athreya	Related

[151] M. Rosenblatt: “Book review, A. M. Yaglom (ed.), ‘Correlation of stationary and random functions’, vols. I and II,” Bull. Am. Math. Soc., New Ser. 20 : 2 (April 1989), pp. 207–211. MR 1567751 article

[152] M. Rosenblatt: “Erratum: ‘Prediction for some non-Gaussian autoregressive schemes’,” Adv. Appl. Math. 10 : 1 (1989), pp. 130. Erratum to an article published in Adv. Appl. Math. 7:2 (1986). MR 977792 Zbl 0663.62104 article

[153] K.-S. Lii and M. Rosenblatt: “Nonminimum phase non-Gaussian deconvolution,” pp. 359–374 in Multivariate statistics and probability: Essays in memory of Paruchuri R. Krishnaiah. Edited by C. R. Rao. 1989. Republished from J. Multivar. Anal. 27:2 (1988). Zbl 0697.62089 incollection

A procedure for deconvolution of nonminimum phase non-Gaussian time series based on the estimation of higher order (greater than two) spectra is given. This can be applied to the analysis of seismograms. The procedure allows estimation of the wavelet. Knowledge of cumulant spectra of order greater than two allows estimation of the phase of the wavelet. In this way one has access to information not available in the ordinary second-order deconvolution procedures. Computational details of the method for estimating the phase of the wavelet are given. There are simulated illustrative examples. One of the examples is based on an actual reflectivity series from a sonic well log. The method is effective asymptotically in the nonminimum phase non-Gaussian context where the Wiener–Levinson procedure does not apply.

Keh-Shin Lii	Related
Paruchuri R. Krishnaiah	Related
Calyampudi Radhakrishna Rao	Related

[154] M. Rosenblatt: “Comments on structure and estimation for non-Gaussian linear processes,” pp. 88–97 in Topics in non-Gaussian signal processing. Edited by E. J. Wegman, S. C. Schwartz, and J. B. Thomas. Springer (New York), 1989. Zbl 0761.62116 incollection

Edward Joseph Wegman	Related
Stuart Carl Schwartz	Related
John Bowman Thomas	Related

[155] F. J. Breidt, R. A. Davis, K.-S. Lii, and M. Rosenblatt: “Nonminimum phase non-Gaussian autoregressive processes,” Proc. Natl. Acad. Sci. U.S.A. 87 : 1 (1990), pp. 179–181. MR 1031950 Zbl 0686.62068 article

F. Jay Breidt	Related
Richard Alan Davis	Related
Keh-Shin Lii	Related

[156] K. S. Lii and M. Rosenblatt: “Asymptotic normality of cumulant spectral estimates,” J. Theor. Probab. 3 : 2 (April 1990), pp. 367–385. MR 1046340 Zbl 0716.62094 article

[157] K. S. Lii and M. Rosenblatt: “Cumulant spectral estimates: Bias and covariance,” pp. 365–405 in Limit theorems in probability and statistics (Pécs, Hungary, 3–7 July 1989). Edited by I. Berkes, E. Csáki, and P. Révész. Colloquia Mathematica Societatis János Bolyai 57. North-Holland (Amsterdam), 1990. MR 1116799 Zbl 0727.62090 incollection

Pál Révész	Related
István Berkes	Related
Keh-Shin Lii	Related
Endre Csáki	Related

[158] F. J. Breidt, R. A. Davis, K.-S. Lii, and M. Rosenblatt: “Maximum likelihood estimation for noncausal autoregressive processes,” J. Multivar. Anal. 36 : 2 (February 1991), pp. 175–198. MR 1096665 Zbl 0711.62072 article

F. Jay Breidt	Related
Richard Alan Davis	Related
Keh-Shin Lii	Related

[159] R. A. Davis and M. Rosenblatt: “Parameter estimation for some time series models without contiguity,” Statist. Probab. Lett. 11 : 6 (1991), pp. 515–521. MR 1116746 Zbl 0725.62079 article

[160] M. Rosenblatt: “Introductory remarks,” pp. viii–x in Spatial stochastic processes: A Festschrift in Honor of Ted Harris on his seventieth birthday. Edited by K. S. Alexander and J. C. Watkins. Progress in Probability 19. Birkhäuser (Boston), 1991. MR 1144087 incollection

Kenneth Sidney Alexander	Related
Theodore Edward Harris	Related
Joseph Clyde Watkins, III	Related

[161] K. Helland, K. S. Lii, and M. Rosenblatt: “Monte Carlo and turbulence,” pp. 405–418 in Nonparametric functional estimation and related topics (Spetses, Greece, 29 July–10 August 1990). Edited by G. Roussas. NATO ASI Series. Series C. Mathematics and Physical Sciences 335. Kluwer Academic (Dordrecht, The Netherlands), 1991. MR 1154342 Zbl 0737.76035 incollection

Keh-Shin Lii	Related
George Gregory Roussas	Related
K. N. Helland	Related

[162] M. Rosenblatt: Stochastic curve estimation. NSF-CBMS Regional Conference Series in Probability and Statistics 3. Institute of Mathematical Statistics (Hayward, CA), 1991. Zbl 1163.62318 book

[163] M. Rosenblatt and B. E. Wahlen: “A nonparametric measure of independence under a hypothesis of independent components,” Stat. Probab. Lett. 15 : 3 (October 1992), pp. 245–252. MR 1190260 Zbl 0770.62039 article

[164] K.-S. Lii and M. Rosenblatt: “An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes,” J. Multivar. Anal. 43 : 2 (November 1992), pp. 272–299. MR 1193615 Zbl 0765.62082 article

[165] New directions in time series analysis (Minneapolis, MN, 2–27 July 1990), Part 1. Edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. S. Taqqu. IMA Volumes in Mathematics and its Applications 45. Springer (New York), 1992. MR 1235574 Zbl 0761.00003 book

John Frederick Geweke	Related
Peter E. Caines	Related
Murad Salman Taqqu	Related
Emanuel Parzen	Related
David Brillinger	Related

[166] M. Rosenblatt: “Gaussian and non-Gaussian linear sequences,” pp. 327–333 in New directions in time series analysis (Minneapolis, MN, 2–27 July 1990), Part 1. Edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. S. Taqqu. IMA Volumes in Mathematics and its Applications 45. Springer (New York), 1992. MR 1235591 Zbl 0850.60011 incollection

John Frederick Geweke	Related
Peter E. Caines	Related
Murad Salman Taqqu	Related
Emanuel Parzen	Related
David Brillinger	Related

[167] New directions in time series analysis (Minneapolis, MN, 2–27 July 1990), Part 2. Edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. S. Taqqu. IMA Volumes in Mathematics and its Applications 45. Springer (New York), 1992. MR 1235595 Zbl 0761.00004 book

John Frederick Geweke	Related
Peter E. Caines	Related
Murad Salman Taqqu	Related
Emanuel Parzen	Related
David Brillinger	Related

[168] M. Rosenblatt: “The central limit theorem and Markov sequences,” pp. 171–177 in Doeblin and modern probability (Blaubeuren, Germany, 2–7 July 1991). Edited by H. Cohn. Contemporary Mathematics 149. American Mathematical Society (Providence, RI), 1993. MR 1229962 Zbl 0787.60080 incollection

Harry Cohn	Related
Wolfgang Doeblin	Related

[169] K.-S. Lii and M. Rosenblatt: “Bispectra and phase of non-Gaussian linear processes,” J. Theor. Probab. 6 : 3 (1993), pp. 579–593. MR 1230347 Zbl 0774.62089 article

The phase of the transfer function of linear processes which cannot be identified in the Gaussian case can be almost fully resolved in the non-Gaussian case. Estimates have been proposed in the past. A nonparametric estimate of the phase with better asymptotic convergence properties as a function of sample size is studied here. The asymptotic behavior of the bias and variance of the estimate is examined. In particular the variance of the phase estimate is shown to be asymptotically independent of the frequency (if the frequency is not zero). Related problems are of interest in deconvolution, transfer function estimation, as well as in the resolution of astronomical images (perturbed by atmospheric turbulence) obtained by earth based telescopes.

[170] K.-S. Lii and M. Rosenblatt: “Non-Gaussian autoregressive moving average processes,” Proc. Natl. Acad. Sci. U.S.A. 90 : 19 (October 1993), pp. 9168–9170. MR 1246981 Zbl 0779.62076 article

[171] M. Kramer and M. Rosenblatt: “The Gaussian log likelihood and stationary sequences,” pp. 69–79 in Developments in time series analysis. Edited by T. Subba Rao. Chapman & Hall (London), 1993. Volume in honour of Maurice B. Priestley. MR 1292259 Zbl 0879.62083 incollection

Maurice Bertram Priestley	Related
Tata Subba Rao	Related
Michael Kramer	Related

[172] M. Rosenblatt: “A note on prediction and an autoregressive sequence,” pp. 291–295 in Stochastic processes: A Festschrift in honour of Gopinath Kallianpur. Edited by S. Cambanis, J. K. Ghosh, R. L. Karandikar, and P. K. Sen. Springer (New York), 1993. MR 1427325 Zbl 0787.60049 incollection

Gopinath Kallianpur	Related
Rajeeva L. Karandikar	Related
Pranab Kumar Sen	Related
Jayanta Kumar Ghosh	Related
Stamatis Cambanis	Related

[173] M. Rosenblatt: “Parameter estimation for some finite parameter stationary sequences,” pp. 213–218 in Probability, statistics and optimisation: A tribute to Peter Whittle. Edited by F. P. Kelly. Wiley Series in Probability and Mathematical Statistics 104. Wiley (Chichester, UK), 1994. MR 1320754 Zbl 0856.62077 incollection

Peter Whittle	Related
Francis Patrick Kelly	Related

[174] M. Rosenblatt: “Prediction and non-Gaussian autoregressive stationary sequences,” Ann. Appl. Probab. 5 : 1 (1995), pp. 239–247. MR 1325051 Zbl 0828.62084 article

[175] K.-S. Lii and M. Rosenblatt: “Maximum likelihood estimation for non-Gaussian nonminimum phase ARMA sequences,” Statist. Sinica 6 : 1 (1996), pp. 1–22. MR 1379046 Zbl 0839.62085 article

We consider an approximate maximum likelihood procedure for estimating parameters of possibly noncausal and noninvertible autoregressive moving average processes driven by independent identically distributed non-Gaussian noise. It is shown that the normalized approximate likelihood has a global maximum at true parameter values in the non-Gaussian case. Under appropriate conditions, estimates of parameters that are solutions of likelihood equations exist, are consistent and asymptotically normal. An asymptotic covariance matrix is given. The procedure is illustrated with simulation examples of \( \mathrm{ARMA}(1,1) \) processes.

[176] K.-S. Lii and M. Rosenblatt: “Nongaussian autoregressive sequences and random fields,” pp. 295–309 in Stochastic modelling in physical oceanography. Edited by R. J. Adler, P. Müller, and B. Rozovskii. Progress in Probability 39. Birkhäuser (Boston), 1996. MR 1383877 Zbl 0865.76075 incollection

In this paper we discuss estimation procedures for the parameters of autoregressive schemes. There is a large literature concerned with estimation in the one-dimensional Gaussian case. Much of our discussion will however be dedicated to the non-Gaussian context, some aspects of which have been considered only in recent years. Results have also at times been obtained in the broader context of autoregressive moving average schemes. We restrict ourselves to the case of autoregressive schemes for the sake of simplicity. Also they are the discrete analogue of simple versions of stochastic differential equations with constant coefficients. It is also apparent that non-Gaussian autoregressive stationary sequences have a richer and more complicated structure than that of the Gaussian autoregressive stationary sequences.

Boris L. Rozovskii	Related
Keh-Shin Lii	Related
Peter Müller	Related
Robert Joseph Adler	Related

[177] M. Rosenblatt: “The likelihood of an autoregressive scheme,” pp. 352–362 in Athens conference on applied probability and time series analysis (Athens, GA, 22–26 March 1995), vol. 2: Time series analysis. Edited by P. M. Robinson and M. Rosenblatt. Lecture Notes in Statistics 115. Springer (New York), 1996. Volume in memory of E. J. Hannan. MR 1466758 incollection

Edward James Hannan	Related
Peter Michael Robinson	Related

[178] Special issue dedicated to Murray Rosenblatt, published as J. Theor. Probab. 10 : 2. Issue edited by A. Mukherjea. Springer (New York), April 1997. Dedicated to Murray Rosenblatt on the occasion of his 70th birthday. MR 1455143 Zbl 0879.00018 book

[179] T. C. Sun: “Murray Rosenblatt: His contributions to probability and statistics,” pp. 279–286 in Special issue dedicated to Murray Rosenblatt, published as J. Theor. Probab. 10 : 2. Issue edited by A. Mukherjea. Springer (New York), April 1997. MR 1455144 Zbl 0895.01013 incollection

Arunava Mukherjea	Related
Tze Chien Sun	Related

[180] “Publications of Murray Rosenblatt,” pp. 287–293 in Special issue dedicated to Murray Rosenblatt, published as J. Theor. Probab. 10 : 2. Issue edited by A. Mukherjea. Springer (New York), April 1997. MR 1455145 incollection

[181] M. Rosenblatt: “Some simple remarks on an autoregressive scheme and an implied problem,” pp. 295–305 in Special issue dedicated to Murray Rosenblatt, published as J. Theoret. Probab. 10 : 2. Issue edited by A. Mukherjea. Springer (New York), April 1997. MR 1455146 Zbl 0895.60039 incollection

[182] M. Rosenblatt: “Comments on estimation and prediction for autoregressive and moving average non-Gaussian sequences,” pp. 353–358 in Stochastic models in geosystems (Minneapolis, MN, 16–20 May 1994). Edited by S. A. Molchanov and W. A. Woyczynski. IMA Volumes in Mathematics and its Applications 85. Springer (Berlin), 1997. MR 1480981 Zbl 0897.62100 incollection

Wojbor A. Woyczynski	Related
S. A. Molchanov	Related

[183] K.-S. Lii and M. Rosenblatt: “Line spectral analysis for harmonizable processes,” Proc. Natl. Acad. Sci. USA 95 : 9 (1998), pp. 4800–4803. MR 1619284 Zbl 0899.62118 article

[184] M. Rosenblatt: “Non-Gaussian autoregressive and moving average schemes,” pp. 731–737 in Asymptotic methods in probability and statistics (Ottawa, 8–13 July 1997). Edited by B. Szyszkowicz. North-Holland (Amsterdam), 1998. A volume in honor of Miklós Csörgő on his 65th birthday. MR 1661514 Zbl 0946.62086 incollection

Consider the systems of equations \[ \sum_{k=0}^p\tilde{\phi}_kx_{t-k} = \sum_{j=0}^q \tilde{\theta}_j\xi_{t-j} \] where \( \xi_t \)’s are independent, identically distributed random variables with \( E\xi_t \equiv O \), \( E\xi_t^2 \equiv 1 \). Let \( \tilde{\phi}_0 = \tilde{\theta}_0 = 1 \) and \( \tilde{\phi}_p, \tilde{\theta}_q \neq 0 \). The polynomials \[ \phi(z) = \sum_{k=0}^p\tilde{\phi}_k z^k \quad\text{and}\quad \theta(z) = \sum_{j=0}^q\tilde{\theta}_j z^j \] are assumed to have no roots in common. There is then a stationary solution \( \{x_t\} \) of the system if and only if \( \phi(z) \) has no roots of absolute value one and this solution is uniqely determined. The solution is called an autoregressive moving average (ARMA) process. If \( \phi(z) \equiv 1 \) the solution is a moving average (MA) while if \( \theta(z) \equiv 1 \) it is an autoregressive scheme (AR). If all the roots of \( \phi(z) \) and \( \theta(z) \) are outside the unit disc \( |z| \leq 1 \) in the complex plane we shall call the solution minimum phase. It is well known that in the case of a stationary ARMA minimum phase process, the best predictor of \( x_t \) in mean square given the past is linear [Shepp et al. 1980]. A necessary condition for the best predictor to be linear is given in terms of a functional equation. This condition is then used in the case of autoregressive schemes and moving averages. A simple example is given of a nonminimum phase scheme in which the best predictor is nonlinear.

Barbara Szyszkowicz	Related
Miklós Csörgő	Related

[185] M. Rosenblatt: Gaussian and non-Gaussian linear time series and random fields. Springer Series in Statistics. Springer, 2000. MR 1742357 Zbl 0933.62082 book

[186] K.-S. Lii and M. Rosenblatt: “Spectral analysis for harmonizable processes,” Ann. Stat. 30 : 1 (2002), pp. 258–297. A correction to this article ws published in Ann. Stat. 31:5 (2003). MR 1892664 Zbl 1012.62099 article

Spectral estimation of nonstationary but harmonizable processes is considered. Given a single realization of the process, periodogram-like and consistent estimators are proposed for spectral mass estimation when the spectral support of the process consists of lines. Such a process can arise in signals of a moving source from array data or multipath signals with Doppler stretch from a single receiver. Such processes also include periodically correlated (or cyclostationary) and almost periodically correlated processes as special cases. We give detailed analysis on aliasing, bias and covariances of various estimators. It is shown that dividing a single long realization of the process into nonoverlapping subsections and then averaging periodogram-like estimates formed from each subsection will not yield meaningful results if one is estimating spectral mass with support on lines with slope not equal to 1. If the slope of a spectral support line is irrational, then spectral masses do not fold on top of each other in estimation even if the data are equally spaced. Simulation examples are given to illustrate various theoretical results.

[187] K.-S. Lii and M. Rosenblatt: “Correction: ‘Spectral analysis for harmonizable processes’,” Ann. Stat. 31 : 5 (2003), pp. 1693. Correction to an article published in Ann. Stat. 30:1 (2002). MR 2012830 article

[188] M. Rosenblatt: “Spectral analysis and a class of nonstationary processes,” pp. 447–450 in Recent advances and trends in nonparametric statistics. Edited by M. G. Akritas and D. N. Politis. Elsevier (Amsterdam), 2003. MR 2562419 incollection

Michael Georgiou Akritas	Related
Dimitris Nicolas Politis	Related

[189] M. Rosenblatt: “Non-Gaussian time series models,” pp. 227–237 in Time series analysis and applications to geophysical systems (Minneapolis, MN, 12–15 November 2001). Edited by D. R. Brillinger, E. A. Robinson, and F. P. Schoenberg. IMA Volumes in Mathematics and its Applications 139. Springer, 2004. MR 2111484 Zbl 1091.62085 incollection

David Brillinger	Related
Frederic Paik Schoenberg	Related
Enders Anthony Robinson	Related

[190] K.-S. Lii and M. Rosenblatt: “Estimation for almost periodic processes,” Ann. Stat. 34 : 3 (2006), pp. 1115–1139. A correction to this article was published in Ann. Stat. 36:3 (2008). MR 2278353 Zbl 1113.62111 article

Processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two-dimensional spectral plane. Methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known. Here methods for estimating the intercepts of the lines of spectral concentration in the Gaussian case are given under appropriate conditions. The methods determine rates of convergence sufficiently fast as the sample size \( n\to\infty \) so that the spectral estimation on the estimated lines can then proceed effectively. This task involves bounding the maximum of an interesting class of non-Gaussian possibly nonstationary processes.

[191] M. Rosenblatt: “An example and transition function equicontinuity,” Statist. Probab. Lett. 76 : 18 (December 2006), pp. 1961–1964. MR 2329240 Zbl 1108.60067 article

[192] F. J. Breidt, R. A. Davis, N.-J. Hsu, and M. Rosenblatt: “Pile-up probabilities for the Laplace likelihood estimator of a non-invertible first order moving average,” pp. 1–19 in Time series and related topics: In memory of Ching-Zong Wei (Taipei, Taiwan, 12–14 December 2005). Edited by H.-C. Ho, C.-K. Ing, and T. L. Lai. IMS Lecture Notes–Monograph Series 52. Institute of Mathematical Statistics (Beachwood, OH), 2006. MR 2427836 Zbl 1268.62107 incollection

The first-order moving average model or \( \mathrm{MA}(1) \) is given by \[ X_t = Z_t - \theta_0 Z_{t-1} ,\] with independent and identically distributed \( \{Z_t\} \). This is arguably the simplest time series model that one can write down. The \( \mathrm{MA}(1) \) with unit root (\( \theta_0 = 1 \)) arises naturally in a variety of time series applications. For example, if an underlying time series consists of a linear trend plus white noise errors, then the differenced series is an \( \mathrm{MA}(1) \) with unit root. In such cases, testing for a unit root of the differenced series is equivalent to testing the adequacy of the trend plus noise model. The unit root problem also arises naturally in a signal plus noise model in which the signal is modeled as a random walk. The differenced series follows a \( \mathrm{MA}(1) \) model and has a unit root if and only if the random walk signal is in fact a constant.

The asymptotic theory of various estimators based on Gaussian likelihood has been developed for the unit root case and nearly unit root case (\( \theta = 1 + \beta/n \), \( \beta\leq 0 \)). Unlike standard \( 1/\sqrt{n} \)-asymptotics, these estimation procedures have \( 1/n \)-asymptotics and a so-called pile-up effect, in which \( \mathrm{P}(\hat{\theta} = 1) \) converges to a positive value. One explanation for this pile-up phenomenon is the lack of identifiability of \( \theta \) in the Gaussian case. That is, the Gaussian likelihood has the same value for the two sets of parameter values \( (\theta,\sigma^2) \) and \( (1/\theta,\theta^2\sigma^2) \). It follows that \( \theta = 1 \) is always a critical point of the likelihood function. In contrast, for non-Gaussian noise, \( \theta \) is identifiable for all real values. Hence it is no longer clear whether or not the same pile-up phenomenon will persist in the non-Gaussian case. In this paper, we focus on limiting pile-up probabilities for estimates of \( \theta_0 \) based on a Laplace likelihood. In some cases, these estimates can be viewed as Least Absolute Deviation (LAD) estimates. Simulation results illustrate the limit theory.

Nan-Jung Hsu	Related
Ching-Kang Ing	Related
F. Jay Breidt	Related
Tze Leung Lai	Related
Richard Alan Davis	Related
Ching Zong Wei	Related
Hwai-Chung Ho	Related

[193] K.-S. Lii and M. Rosenblatt: “Correction: ‘Estimation for almost periodic processes’,” Ann. Stat. 36 : 3 (2008), pp. 1508. Correction to an article published in Ann. Stat. 34:3 (2006). MR 2418665 article

[194] K. S. Lii and M. Rosenblatt: “Prolate spheroidal spectral estimates,” Statist. Probab. Lett. 78 : 11 (August 2008), pp. 1339–1348. MR 2444324 Zbl 1144.62079 article

An estimate of the spectral density of a stationary time series can be obtained by taking the finite Fourier transform of an observed sequence \( x_0 \), \( x_1,\dots \), \( x_{N-1} \) of sample size \( N \) with taper a discrete prolate spheroidal sequence and computing its square modulus. It is typical to take the average \( K \) of several such estimates corresponding to different prolate spheroidal sequences with the same bandwidth \( W(N) \) as the final computed estimate. For the mean square error of such an estimate to converge to zero as \( N\to\infty \), it is shown that it is necessary to have \( W(N)\downarrow 0 \) with \[ NW(N)\to\infty \] as \( N\to\infty \) and significantly have \[ K(N) \leq 2NW(N) \] but \( K = K(N)\to\infty \) as \( N\to\infty \).

[195] U. Grenander and M. Rosenblatt: Statistical analysis of stationary time series, 2nd, republished edition. AMS Chelsea Publishing Series 320. American Mathematical Society (Providence, RI), 2008. Zbl 0902.62115 book

[196] M. Rosenblatt: “A comment on a conjecture of N. Wiener,” Statist. Probab. Lett. 79 : 3 (February 2009), pp. 347–348. MR 2493017 Zbl 1165.60016 article

[197] D. R. Brillinger and R. A. Davis: “A conversation with Murray Rosenblatt,” Stat. Sci. 24 : 1 (2009), pp. 116–140. MR 2561129 Zbl 1327.01036 article

David Brillinger	Related
Richard Alan Davis	Related

[198] M. Rosenblatt: “Spectral analysis for processes with almost periodic covariances,” J. Stat. Plann. Inference 140 : 12 (December 2010), pp. 3608–3612. MR 2674150 Zbl 1233.62166 article

[199] M. Rosenblatt: “Stationary processes and a one-sided representation in terms of independent identically distributed random variables,” pp. 311–315 in Dependence in probability, analysis and number theory (Graz, Austria, 17–20 June 2009). Edited by I. Berkes, R. C. Bradley, H. Dehling, M. Peligrad, and R. Tichy. Kendrick Press (Heber City, UT), 2010. Volume in honor of Walter Philipp. This paper was dedicated to Philipp and the author’s deceased wife. MR 2731063 Zbl 1213.60070 incollection

István Berkes	Related
Richard Crane Bradley, Jr.	Related
Robert Franz Tichy	Related
Herold G. Dehling	Related
Magda Peligrad	Related
Walter Philipp	Related

[200] M. Rosenblatt: Selected works of Murray Rosenblatt. Edited by R. A. Davis, K.-S. Lii, and D. N. Politis. Selected Works in Probability and Statistics. Springer (Berlin), 2011. MR 2742596 Zbl 1232.60004 book

Dimitris Nicolas Politis	Related
Richard Alan Davis	Related
Keh-Shin Lii	Related

[201] M. Rosenblatt: “Remarks suggested by the paper of H. Tong,” Stat. Interface 4 : 2 (2011), pp. 121–122. MR 2812804 Zbl 05983885 article

[202] K.-S. Lii and M. Rosenblatt: “Estimation for a class of nonstationary processes,” Statist. Probab. Lett. 81 : 11 (November 2011), pp. 1612–1622. MR 2832920 Zbl 1227.62077 article

Random processes with almost periodic covariance function are considered from a spectral outlook. Given suitable conditions, spectral estimation problems are discussed for Gaussian processes of this type that are neither stationary nor locally stationary. Spectral mass is concentrated on lines parallel to the main diagonal in the spectral plane. A method of estimation of the support of spectral mass under appropriate restraints is considered. Some open questions are discussed. Extension of the methods for a class of non-Gaussian nonstationary processes with mean value function a trigonometric regression is given. Consistent estimates for frequency, amplitude and phase of the regression are noted when the residual process is zero mean almost periodic. The resulting estimation of the spectral mass of the residual is also considered.

[203]M. Bhattacharjee, R. Lockhart, J. Rolph, G. Roussas, H. Tucker, R. J.-B. Wets, P. Bickel, T. S. Ferguson, A. Lo, M. L. Puri, S. Stigler, W. Sudderth, Y. Yatracos, D. Brillinger, L. A. Goodman, J. Shaffer, H. Chernoff, P. Diaconis, M. Rosenblatt, and F. J. Samaniego: “A tribute to David Blackwell.” Edited by G. G. Roussas. Notices Am. Math. Soc. 58 : 7 (2011), pp. 912–928. MR 2850553 Zbl 1225.01082 article

D. H. Blackwell	Related	Volume
Thomas S. Ferguson	Related
Leo A. Goodman	Related
Albert Yee-Lap Lo	Related
George Gregory Roussas	Related
Roger J-B Wets	Related
Madan Lal Puri	Related
Francisco J. Samaniego	Related
Juliet Popper Shaffer	Related
John Rolph	Related
Manish C. Bhattacharjee	Related
Stephen Mack Stigler	Related
William D. Sudderth	Related
Yannis Yatracos	Related
Richard Lockhart	Related
Howard Tucker	Related
Peter J. Bickel	Related
David Brillinger	Related
Herman Chernoff	Related
Persi Diaconis	Related

@article {key2850553m,
	AUTHOR = {Bhattacharjee, Manish and Lockhart,
		 Richard and Rolph, John and Roussas,
		 George and Tucker, Howard and Wets,
		 Roger J.-B. and Bickel, Peter and Ferguson,
		 Thomas S. and Lo, Albert and Puri, Madan
		 L. and Stigler, Stephen and Sudderth,
		 W. and Yatracos, Yannis and Brillinger,
		 David and Goodman, Leo A. and Shaffer,
		 Juliet and Chernoff, Herman and Diaconis,
		 Persi and Rosenblatt, Murray and Samaniego,
		 Francisco J.},
	TITLE = {A tribute to {D}avid {B}lackwell},
	JOURNAL = {Notices Am. Math. Soc.},
	FJOURNAL = {Notices of the American Mathematical
		 Society},
	VOLUME = {58},
	NUMBER = {7},
	YEAR = {2011},
	PAGES = {912--928},
	URL = {http://www.ams.org/notices/201107/rtx110700912p.pdf},
	NOTE = {Edited by G. G. Roussas.
		 MR:2850553. Zbl:1225.01082.},
	ISSN = {0002-9920},
	CODEN = {AMNOAN},
}

[204] M. Rosenblatt: “Short range and long range dependence,” pp. 283–294 in Asymptotic laws and methods in stochastics (Ottawa, 3–6 July 2012). Edited by D. Dawson, R. Kulik, M. Ould Haye, B. Szyszkowicz, and Y. Zhao. Fields Institute Communications 76. Fields Institute (Toronto), 2015. A volume in honour of Miklós Csörgő on the occasion of his 80th birthday. MR 3409836 Zbl 1368.60038 incollection

Rafał Kulik	Related
Barbara Szyszkowicz	Related
Miklós Csörgő	Related
Donald Dawson	Related
Yiqiang Q. Zhao	Related
Mohamedou Ould Haye	Related

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