By G. M. Feldman

ISBN-10: 0821845934

ISBN-13: 9780821845936

This ebook reports the matter of the decomposition of a given random variable right into a sum of self sufficient random variables (components). ranging from the recognized Cramér theorem, which says that every one elements of a typical random variable also are general random variables, the crucial characteristic of the booklet is Fel'dman's use of strong analytical options. within the algebraic case, one can't at once use analytic tools as a result absence of a common analytic constitution at the twin staff, that's the area of attribute services. however, the tools constructed during this ebook let one to use analytic thoughts within the algebraic environment. the 1st a part of the ebook provides effects at the mathematics of chance distributions of random variables with values in a in the neighborhood compact abelian team. the second one half reviews difficulties of characterization of a Gaussian distribution of a in the community compact abelian staff via the independence or exact distribution of its linear statistics.

Readership: experts in chance idea, mathematical data and practical research.

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

**Example text**

The structure of Gaussian distributions on such groups is well known. If u E IF s (X) , then either ,u = ,uac or ,u = ,us . 6. Indeed, set G = U (v) . Then, if p (G) = X , then µ = ,uac . On the other hand, if p (G) 54 X, then Rn Now let dim X = oo. Then X --+ T °O . We need the Kakutani theorem [Ka], which is stated as follows. THEOREM. 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.

Let X be a connected group of finite dimension 1. Then any two Gaussian distributions on X are either mutually absolutely continuous or mutually singular. PROOF. 15, X Rn + K, where n > 0 and K is a connected compact group. To avoid complicated notation, we restrict ourselves to considering the case X = K , dim K = l . Put D = K* . 6. Then Z1 c f(D) c f(D). 20 yields ker p = A (Tf(D)) c A (RZ') Z1. Put G = ker p . Let first #1 , ,u2 E s (K) . 6, p = p (v1) , where vi E f f s (R1) , i = 1 , 2 . Put Li = U (v1) , i = 1 , 2 .

REMARK. 1(ii). Then (P (Y2) = 0 for any y2 E Yo and p(yi + y2) = p(yi) for any Y j E Y, yZ E Yo . Indeed, let Yi be a compact subgroup of Y. 1(ii) over Yj with respect to the measure dmYI (y1), we obtain p(y2) = 0 for y2EY2. Let yj E Y, y2 E Yo, and let MYZ be the closed subgroup generated by y2. Consider the function P(l) = rp(yl + lye) on the group Z. 1(ii) that the function P(l) satisfies the equation P(l + 3) 3P(1 + 2) + 3P(l + 1) - P(l) = 0. The latter implies (see [Ge, Chapter V, §3]) that P(1) = aol2 + all + a2 where the aj depend on yj and y2 .

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

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