# New PDF release: Arithmetic of Probability Distributions, and

By G. M. Feldman

This e-book stories 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 parts of an ordinary random variable also are basic random variables, the imperative function of the publication is Fel'dman's use of strong analytical recommendations. within the algebraic case, one can't without delay use analytic tools as a result absence of a usual analytic constitution at the twin crew, that's the area of attribute services. however, the tools constructed during this publication enable one to use analytic thoughts within the algebraic atmosphere. the 1st a part of the ebook provides effects at the mathematics of likelihood distributions of random variables with values in a in the neighborhood compact abelian staff. the second one half stories difficulties of characterization of a Gaussian distribution of a in the community compact abelian team via the independence or exact distribution of its linear statistics.

Readership: experts in chance conception, mathematical records and useful research.

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

**Example text**

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. Then P (,U) E I, S (Rn + T m) , p (u) (p (Uo)) = 0 , and p(p)(') = 1 only at ' = 0, ' E ]n + Zm , but this is impossible since the set p (Uo) is open. The contradiction we have come to shows that u (U) 0. El Let us prove now that the "linearization" is possible for an arbitrary symmetric Gaussian distribution.

Bnsn + ... )2} is a divisor of the distribution v0 . Apparently, the support a (v1) = L = It E . Note that the distribution ul = p1(v1) E I,s(K) is a divisor of u . The support of the distribution ul is the subgroup p1 (L) ti T , since the kernel kerp1 contains (as was proved. above) no subgroups topologically isomorphic to R. 20, the distribution u and, R°° : t = Ab , A E R} 1 hence, ,u has non-Gaussian divisors. The contradiction obtained implies that kerp1 n l2 = {0}. §5. GAUSSIAN DISTRIBUTION 51 Now we can construct the derived subspace G.

10 enables us to prove this alternative for finite-dimensional groups and for some class of infinite-dimensional groups. 0 0 0 II. 16. PROPOSITION. 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).