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 Diskr. Mat., 1999, Volume 11, Issue 4, Pages 48–57 (Mi dm391)

Conditions for the uniqueness of the moment problem in the class of $q$-distributions

A. N. Alekseichuk

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.

DOI: https://doi.org/10.4213/dm391

Full text: PDF file (658 kB)

English version:
Discrete Mathematics and Applications, 1999, 9:6, 615–625

Bibliographic databases:

UDC: 519.21

Citation: A. N. Alekseichuk, “Conditions for the uniqueness of the moment problem in the class of $q$-distributions”, Diskr. Mat., 11:4 (1999), 48–57; Discrete Math. Appl., 9:6 (1999), 615–625

Citation in format AMSBIB
\Bibitem{Ale99} \by A.~N.~Alekseichuk \paper Conditions for the uniqueness of the moment problem in the class of $q$-distributions \jour Diskr. Mat. \yr 1999 \vol 11 \issue 4 \pages 48--57 \mathnet{http://mi.mathnet.ru/dm391} \crossref{https://doi.org/10.4213/dm391} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1761011} \zmath{https://zbmath.org/?q=an:0971.05015} \transl \jour Discrete Math. Appl. \yr 1999 \vol 9 \issue 6 \pages 615--625 

• http://mi.mathnet.ru/eng/dm391
• https://doi.org/10.4213/dm391
• http://mi.mathnet.ru/eng/dm/v11/i4/p48

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This publication is cited in the following articles:
1. Cooper C., “On the distribution of rank of a random matrix over a finite field”, Random Structures & Algorithms, 17:3–4 (2000), 197–212
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