Celebratio Mathematica

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

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

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

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.

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

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

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Murray Rosenblatt

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