On the property of decomposability of functions of k-valued logic related to summation of n-dependent random variables in a finite Abelian group
Igor Aleksandrovich Kruglov · Discrete Mathematics and Applications · 2005
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 (ξ 1 , . . . ,ξ n ) ⊕ f (ξ 2 , . . . ,ξ n +1 ) ⊕ . . . ⊕ f (ξ N , . . . ,ξ N + n −1 ), where ξ 1 ,ξ 2 , . . . 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 ( x 1 , . . . , x n ) ⊖ f (0, . . . ,0) ⊕ H = g ( x 1 , . . . , x n −1 ) ⊖ g ( x 2 , . . . , 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 .