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 Diskr. Mat.: Year: Volume: Issue: Page: Find

 Diskr. Mat., 2016, Volume 28, Issue 3, Pages 14–25 (Mi dm1380)

Asymptotics of conditional probabilities of succesful allocation of random number of particles into cells

A. I. Afonina, I. R. Kayumov, A. N. Chuprunov

Kazan (Volga Region) Federal University

Abstract: The article is devoted to the memory of Valentin Fedorovich Kolchin.\qquad\qquad\qquad\qquad\qquad\linebreak Let $\zeta$, $\zeta_i$ ($i\inN$) be independent identically distributed nonnegative integer-valued random variables, $(\eta_{i1},…, \eta_{iN})$ be the fillings of cells in the generalized scheme of allocation of $\zeta_i$ particles into $N$ cells, $1\le i\le n$, for fixed $Z_n=(\zeta_1,\ldots,\zeta_n)$ these allocation schemes are independent. We consider the conditional probabilities $P(A_{n, N} |Ż_n)$ of the event\linebreak $A_{n, N}=\{each cell in each of n allocation schemes contains no more than r particles\}$, where $r$ is some fixed number. The sufficient conditions for the convergence of the sequence $P(A_{n, N} |Ż_n)$ to a nonrandom limit with probability 1 are given. It is shown that the random variable $\ln P(A_{n, N} |Ż_n)$ is asymptotically normal. Applications of the obtained results to the noise-proof encoding are discussed.

Keywords: generalized allocation scheme, Cauchy integral, Hamming code.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-0035115-41-02433 This work was financially supported by the Russian Foundation for Basic Research, project No 14-01-00351, and by the RFBR and the Government of the Republic of Tatarstan, the project No 15-41-02433.

DOI: https://doi.org/10.4213/dm1380

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English version:
Discrete Mathematics and Applications, 2017, 27:5, 277–286

Bibliographic databases:

UDC: 519.212.2
Revised: 26.07.2016

Citation: A. I. Afonina, I. R. Kayumov, A. N. Chuprunov, “Asymptotics of conditional probabilities of succesful allocation of random number of particles into cells”, Diskr. Mat., 28:3 (2016), 14–25; Discrete Math. Appl., 27:5 (2017), 277–286

Citation in format AMSBIB
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