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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
Abstract:
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.
Received: 15.02.2005
Citation:
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”, Diskr. Mat., 17:4 (2005), 29–39; Discrete Math. Appl., 15:5 (2005), 463–473
Linking options:
https://www.mathnet.ru/eng/dm127https://doi.org/10.4213/dm127 https://www.mathnet.ru/eng/dm/v17/i4/p29
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