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Central Limit Theorems for Interchangeable Processes

Published online by Cambridge University Press:  20 November 2018

J. R. Blum,
H. Chernoff,
M. Rosenblatt and
H. Teicher
J. R. Blum
Affiliation:
Indiana University
H. Chernoff
Affiliation:
Stanford University
M. Rosenblatt
Affiliation:
Purdue University
H. Teicher
Affiliation:
Purdue University
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Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i1, i2, H3 … , ik, the joint distribution of

depends merely on k and is independent of the integersi1, i2, … , ik. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 10 , 1958, pp. 222 - 229
Copyright
Copyright © Canadian Mathematical Society 1958 

Footnotes

Initially this problem was investigated by two pairs of authors: Blum and Rosenblatt on the one hand, and Chernoff and Teicher on the other. In view of the great overlap in results, a single combined paper was arranged.

Professor Blum's work was sponsored by the Office of Ordnance Research, U.S. Army under Contract DA-33-008-ORD-965; that of the other three authors was sponsored by the Office of Naval Research.

References

1. Berry, A. C., The accuracy of the Gaussian approximation to the sum of independent variâtes, Trans. Amer. Math. Soc, 49 (1941), 122-136.Google Scholar
2. Cramer, Harold, Random variables and probability distributions, Cambridge Tracts in Mathematics, 136 (Cambridge, 1937).Google Scholar
3. Finetti, Bruno De, La prévision, ses lois logiques, ses sources subjectives, Annales de l'Institut Henri Poincaré, 7 (1937), 1-68.Google Scholar
4. Esseen, Carl-Gustav, Fourier analysis of distribution functions, Acta Math., 77 (1944), 1–125.Google Scholar