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I. A. Kruglov

On the property of decomposability of functions of k-valued logic related to summation of n-dependent random variables in a finite Abelian group

In this paper, we study the limit behaviour of the sequence of distributions of random variables taking values in the finite Abelian group (Ω, ⊕), Ω = {0, 1, . . . ,k − 1}, which admit the representation.

η^(N) = f(ξ₁, . . . ,ξ_n ) ⊕ f(ξ₂, . . . ,ξ _n+1) ⊕ . . . ⊕ f(ξ_N, . . . ,ξ_N+n−1),

where ξ₁,ξ₂, . . . is the initial sequence of independent identically distributed random variables which take values in Ω, f is a k-valued function of n variables which takes values in Ω. We show that the limit behaviour of the sequence of distributions of η^N as N → ∞ is determined by the minimal subgroup H of the group (Ω, ⊕) which for all x ₁, . . . ,x_n ∈ Ω admits the expansion

f(x₁, . . . ,x _n) ⊖ f(0, . . . ,0) ⊕ H = g(x₁, . . . ,x_n−1) ⊖ g(x₂, . . . ,x_n ) ⊕ H

with some k-valued function g of n − 1 variables, where ⊖ is the subtraction operation in the group (Ω, ⊕). We give a description of the limit points of the sequence of distributions of the random variables η^N and converging to them sequences in terms of the subgroup H and the corresponding function g.

Discrete Mathematics and Applications, Walter de Gruyter

Print ISSN: 0924-9266
Volume: 15, 10/2005
Seiten: 463 - 473

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