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By G. M. Feldman

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 well-known Cramér theorem, which says that every one parts of a typical random variable also are common random variables, the principal function of the booklet is Fel'dman's use of strong analytical ideas. within the algebraic case, one can't without delay use analytic equipment as a result of absence of a average analytic constitution at the twin crew, that is the area of attribute features. however, the tools built during this ebook enable one to use analytic recommendations within the algebraic atmosphere. the 1st a part of the e-book provides effects at the mathematics of chance distributions of random variables with values in a in the community compact abelian workforce. the second one half reviews difficulties of characterization of a Gaussian distribution of a in the neighborhood compact abelian crew by way of the independence or exact distribution of its linear statistics.

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

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

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The function (h), h E H, equals 1 at h = 0 only. 10, the group G is connected. Suppose that µ(U) = 0 for some open set U c G, and let Uo be an open subset of U such that Uo + V c U for some neighborhood V of II. ARITHMETIC OF DISTRIBUTIONS 36 zero in G. The group G is connected. 19 the neighborhood V contains a subgroup K such that the group G/K is topologically isomorphic to Rn + Tm for some n, m > 0. Denote this isomorphism by , and put p = TO Tc , where Tc : G - G/K is the natural homomorphism.

Any distribution It on a group X can be represented as the convolution µ = mK * A, where A is a distribution without nondegenerate idempotent divisors and mK is the maximal idempotent divisor of It. PROOF. Let E = {y E Y: ,u(y) :A 01, and let H be the subgroup gener- ated by the set E. 7, its annihilator K = A(X, H) is compact. The distribution mK is just the required maximal idempotent divisor of y. Let T: X -, X/K be the natural homomorphism. 10(h), the restriction to H of the characteristic function µ(y) is the characteristic function of the distribution v = T(µ) E dl '(X/K) .

Necessity. 1 5 implies X Ian + K , where n >0 and K is a connected compact group. Then Y Ian + D, where D = K*. Let the group X be topologically nonisomorphic to a group f the form (i). Then K T. Assume dim K = oo (if dim K < oo , then the reasoning would be obviously simplified). 9. Choose an independent set of real numbers {ak}, set a = (a1 , ... ) E R°° , and consider a continuous homomorphism f1: Y Rn+1 defined by the formula f1(s1, ... , sn ; d) = (s1 , ... , sn ; (f(d), a)). Put pi = fl; then p,: Rn+ 1 -* X.

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