Abstract:
Let $K_q$ be the class of probability distributions on the set
of non-negative integer powers of a number $q>1$ ($q$-distributions),
$\mathsf P=\{p_k=P(q^k),\ k=0,1,\ldots\}$ is a distribution from the class
$K_q$ which has the moments of all orders. It is shown that in order that
the distribution $\mathsf P$ is uniquely determined in the class $K_q$
by the sequence of its moments provided that
$p_k>0$, $k=0,1,\ldots$, it is necessary, and under the condition that
$$
\operatornamewithlimits{sup\,lim}_{k\to\infty} (p_kq^{\binom k2})^{1/k}<\infty,
$$
sufficient, that
$$
\operatornamewithlimits{inf\,lim}_{k\to\infty} p_{2k}q^{\binom{2k}k}
=\operatornamewithlimits{inf\,lim}_{k\to\infty} p_{2k+1}q^{\binom{2k+1}{2}}=0.
$$
These results are applied in the study
of the limit distribution of the number of solutions of a system of random
homogeneous equations with equiprobable matrix of coefficients over
a finite local ring of principle ideals.
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