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

I. A. Kruglov

Diskretnaya Matematika, Tome 17 (2005) no. 4, pp. 29-39.

Résumé

In this paper, we study the limit behaviour of the sequence of distributions of random variables taking values in the finite Abelian group $(\Omega,\oplus)$, $\Omega=\{0,1,\dots,k-1\}$, which admit the representation $$ \eta^{(N)}=f(\xi_1,\dots,\xi_n)\oplus f(\xi_2,\dots,\xi_{n+1}) \oplus\ldots \oplus f(\xi_N,\dots,\xi_{N+n-1}), $$ where $\xi_1,\xi_2,\dotsc$ is the initial sequence of independent identically distributed random variables which take values in $\Omega$, $f$ is a $k$-valued function of $n$ variables which takes values in $\Omega$. We show that the limit behaviour of the sequence of distributions of $\eta^{(N)}$ as $N\to\infty$ is determined by the minimal subgroup $H$ of the group $(\Omega,\oplus)$ which for all $x_1,\dots,x_n\in \Omega$ admits the expansion $$ f(x_1,\dots,x_n)\ominus f(0,\dots,0)\oplus H= g(x_1,\dots,x_{n-1})\ominus g(x_2,\dots,x_n)\oplus H $$ with some $k$-valued function $g$ of $n-1$ variables, where $\ominus$ is the subtraction operation in the group $(\Omega,\oplus)$. We give a description of the limit points of the sequence of distributions of the random variables $\eta^{(N)}$ and converging to them sequences in terms of the subgroup $H$ and the corresponding function $g$. This research was supported by the Program of the President of the Russian Federation for support of leading scientific schools, grant 2358.2003.9.

Bibliographie

[1] Gorchinskii Yu. N., Kruglov I. A., Kapitonov V. M., “Voprosy teorii raspredelenii na konechnykh gruppakh”, Trudy po diskretnoi matematike, 1, 1997, 85–112 | MR | Zbl