By G. M. Feldman

ISBN-10: 0821845934

ISBN-13: 9780821845936

This publication reviews the matter of the decomposition of a given random variable right into a sum of self sustaining random variables (components). ranging from the recognized Cramér theorem, which says that each one parts of a typical random variable also are general random variables, the principal function of the e-book is Fel'dman's use of robust analytical ideas. within the algebraic case, one can't at once use analytic tools as a result absence of a common analytic constitution at the twin team, that is the area of attribute features. however, the equipment built during this publication let one to use analytic thoughts within the algebraic surroundings. the 1st a part of the booklet provides effects at the mathematics of chance distributions of random variables with values in a in the neighborhood compact abelian workforce. the second one half stories difficulties of characterization of a Gaussian distribution of a in the community compact abelian crew by way of the independence or exact distribution of its linear statistics.

Readership: experts in chance thought, mathematical facts and sensible research.

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**Additional info for Arithmetic of Probability Distributions, and Characterization Problems on Abelian Groups**

**Sample text**

Denote by { an element of order 2 in K, and rewrite (1) in the form y(E) = I [A(E) + A(E + E E R(X). Obviously, y(E) = y(E + C) (2) for any E E R (X) . The following three cases are possible, depending on the structure of the support Q(µ) of the distribution It . 1. The support a(y) consists of two points. Taking (2) into account, it can easily be checked that this is possible if and only if µ = mK * Ex. 2. The support a(y) consists of four points. Let Q(µ) _ {x1 , x2 , x3 , x4} . Without loss of generality, we may suppose that xi = 0 and xj+2 = xi + , µ{xj } = µ{xj+2} j = 1 , 2.

Does not contain a subgroup isomorphic to T. Then any two Gaussian distributions on X are either mutually absolutely continuous or mutually singular. PROOF. 15, X Pzft-o Ian + K , where n >0 and K is a con- nected compact group. 16, we restrict ourselves to the case X = K , dim K = oo. Put D = K* . 9. 17, f(D) = R . But then, as can easily be seen kerp = {0}, and, hence, the homomorphism p realizes a one-to-one correspondence between the distributions on R°° and the distributions on K , concentrated on p(R°°) .

So, kerp = S 7L`n . Let there exist y0 E G n kerp . We may assume that the element yo is chosen in such a way that Ay0 V G n kerp for all A E]0 , 1 [. ) defines an inner product on the factorspace II81 /kerA, and yo) is a linear functional on this space. It follows from inequality (1) that the distribution v1 E I'(Il8l) having the characteristic function va(s) = exp{-e(s, yo)2}, s E R1, (2) is a divisor of v. Consequently, the distribution µl = p(vi) is a divisor of u. , H R. By construction p(H) T.

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

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