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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1968, Volume 13, Issue 2, Pages 295–307 (Mi tvp846)

Necessary and sufficient convergence conditions for the convolution of non-identical distributions given on a finite group

V. M. Maksimov

Moscow

Abstract: Say that the convolution of a sequence of distributions $x_1,x_2,…,x_n,…$ on a finite group $G$ converges if, for all $i=1,2,…,$ the sequences $x_ix_{i+1}…x_{i+n}$ converge as $n\to\infty$, each $x_i$ being viewed as the element ${p_1}^ie_1+…+{p_s}^ie_s$ of the algebra over the field of real numbers with the basis $e_1,…,e_s\in G$, where ${p_k}^i$ is the probability of $e_k$ given by $x_i$.
In the paper the necessary and sufficient conditions of such a convergence are found. In particular, the necessary and sufficient conditions are obtained that $\{x_i…x_{i+n}\}$, $i=1,2,…$, converge to the uniform distribution on $G$.

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English version:
Theory of Probability and its Applications, 1968, 13:2, 287–298

Bibliographic databases:

Citation: V. M. Maksimov, “Necessary and sufficient convergence conditions for the convolution of non-identical distributions given on a finite group”, Teor. Veroyatnost. i Primenen., 13:2 (1968), 295–307; Theory Probab. Appl., 13:2 (1968), 287–298

Citation in format AMSBIB
\Bibitem{Mak68} \by V.~M.~Maksimov \paper Necessary and sufficient convergence conditions for the convolution of non-identical distributions given on a~finite group \jour Teor. Veroyatnost. i Primenen. \yr 1968 \vol 13 \issue 2 \pages 295--307 \mathnet{http://mi.mathnet.ru/tvp846} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=230342} \zmath{https://zbmath.org/?q=an:0239.60010|0192.55402} \transl \jour Theory Probab. Appl. \yr 1968 \vol 13 \issue 2 \pages 287--298 \crossref{https://doi.org/10.1137/1113032}