By G. M. Feldman

This booklet 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 every one parts of a standard random variable also are common random variables, the crucial characteristic of the booklet is Fel'dman's use of strong analytical recommendations. within the algebraic case, one can't at once use analytic tools as a result of absence of a ordinary analytic constitution at the twin crew, that's the area of attribute services. however, the equipment built during this booklet enable one to use analytic concepts within the algebraic atmosphere. the 1st a part of the e-book offers effects at the mathematics of likelihood distributions of random variables with values in a in the neighborhood compact abelian team. the second one half experiences difficulties of characterization of a Gaussian distribution of a in the neighborhood compact abelian staff by means of the independence or exact distribution of its linear statistics.

Readership: experts in likelihood conception, mathematical information and useful research.

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**Sample text**

Has no nondegenerate idempotent divisors. Let us show that one can choose numbers ai , j = 1, 2, 3, not simultaneously equal to zero, such that the equality (y1) = 0 holds for some Y, E t + H. Then A must have an indecomposable divisor. Indeed, by the construction, . has no nondegener- ate idempotent divisors. 3, . 17. Let y 1 E i + H. It is clear that there exist a , (x, Yi)d(x) VXi -1

17. PROPOSITION. 19. 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°°) .