By G. M. Feldman

ISBN-10: 0821845934

ISBN-13: 9780821845936

This publication reports the matter of the decomposition of a given random variable right into a sum of autonomous random variables (components). ranging from the recognized Cramér theorem, which says that every one elements of a regular random variable also are common random variables, the relevant function of the e-book is Fel'dman's use of robust analytical options. within the algebraic case, one can't at once use analytic tools as a result absence of a average analytic constitution at the twin team, that's the area of attribute capabilities. however, the tools built during this e-book enable one to use analytic concepts within the algebraic surroundings. the 1st a part of the ebook offers effects at the mathematics of likelihood distributions of random variables with values in a in the community compact abelian team. the second one half stories difficulties of characterization of a Gaussian distribution of a in the community compact abelian team through the independence or exact distribution of its linear statistics.

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**Sample text**

Then either ,u = mK * EX , where K C X and K ti 7L(2) or ,u has an indecomposable divisor. 6. PROOF. 1 such a divisor exists). Assume first that K 96 Z(2). It follows from the previous section that one can choose an indecomposable distribution on K. Any such distribution is an indecomposable divisor of ,u . It remains to consider the case K ti Z(2). 1, the decomposition ,u = mK * AO takes place, where Ao has no nondegenerate divisors. Assuming that ,u 54 m K* EX , let us show that a decomposition /2 = mK * A, (1) exists, in which the distribution A is either indecomposable itself or has an indecomposable divisor.

Let {,uk } and {Vk } be two sequences of distributions on a space n. Assume that for all k = 1, 2 , 3 , ... the distributions ,uk and Vk are mutually absolutely continuous. Then the direct products of the distributions 00 00 ,u=®/2k, v =®Vk k=1 k=1 are mutually absolutely continuous if the product 00 k=1 ldvk V Yyk converges (dvk/d/lk is the Radon-Nikodym derivative), and mutually singular if this product diverges. To simplify notation, we suppose that X = T' . 9) is diagonal A = diag{ 2 a 2,5 5 2 ok , ...

20 implies that u 1 and, hence, 1 ,u have non-Gaussian divisors. The case of a nonsymmetric distribution y can easily be reduced to the considered one. 24. COROLLARY. For every Gaussian distribution on a group X to have non-Gaussian divisors, it is necessary and sufficient that the group X be topologically isomorphic to T. 25. REMARK. 12 such that o(µ) = X. 23 give rise to another construction of such a distribution. 26. REMARK. Let µ E I'S(T) . 20 implies that ,u has a non-Gaussian divisor. But one may assert even more.

### Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups by G. M. Feldman

by Thomas

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