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Conditions for the limit distribution equiprobability in a linear autoregression scheme with random control on a finite group
I. A. Kruglov
Abstract:
We consider the sequence of random variables
$$
\mu^{(N)}=\xi_N(\mu^{(N-1)})^{\zeta_N},
\qquad
N=1,2,\dots,
$$
where
$\mu^{(0)}$ is a random variable that takes values in a finite group
$G=(G, \bullet)$,
$(\xi_N, \zeta_N)$,
$N=1,2,\dots$,
is a sequence of identically distributed random variables that take values
in the Cartesian product
$G\times\operatorname{Aut}G$, where
$(\operatorname{Aut}G, \circ)$
is the group of automorphisms of
$G$. We assume that the random variables
$\mu^{(0)}$,
$(\xi_N, \zeta_N)$,
$N=1,2,\dots$, are independent.
Given an arbitrary distribution of
$\mu^{(0)}$,
we find general necessary and sufficient conditions for the convergence, as
$N\to\infty$,
of the sequence of distributions of random variables
$\mu^{(N)}$
to the equiprobable on
$G$ distribution.
This research was supported by the Program of the President of the Russian Federation
for supporting the leading scientific schools, grant 2358.2003.9.
UDC:
519.2 Received: 15.12.2004
DOI:
10.4213/dm112
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