Celebratio Mathematica

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

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

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.

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]).

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

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

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

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

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].

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.

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

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

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].

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

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.

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

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

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

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

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

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

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

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

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.

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

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Murray Rosenblatt

A conversation with Murray Rosenblatt

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