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 Diskr. Mat., 2017, Volume 29, Issue 2, Pages 109–132 (Mi dm1422)

Local limit theorems for one class of distributions in probabilistic combinatorics

A. N. Timashev

Institute of Cryptography, Communications and Informatics, Academy of Federal Security Service of Russian Federation, Moscow

Abstract: Let a function $f(z)$ be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius $R.$ Let the distribution of the random variable $\xi_n$, $n\in\{1,2,\ldots\}$, be defined by the formula
$$P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}(\frac{(f(z))^N}{N!})}{\mathrm{coeff}_{z^n}(\exp(f(z)))}, N=0,1,\ldots$$
for some $|z|<R$ (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of $\xi_n$ are proved in two cases: a) if $f(z) = (1-z)^{-\la}, \la = \mathrm {const} \in(0,1]$ for $|z| <1$, and b) if all positive coefficients of expansion $f (z)$ in a power series are equal to 1 and the set $A$ of their numbers has the form
$$A = \{m^r | m \in \mathbb{N} \}, r = \mathrm {const},\; r \in \{2,3,\ldots\}.$$
A hypothetical general local limit normal theorem for random variables $\xi_n$ is stated. Some examples of validity of the statement of this theorem are given.

Keywords: power series distributions, local asymptotical normality.

DOI: https://doi.org/10.4213/dm1422

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English version:
Discrete Mathematics and Applications, 2018, 28:6, 405–420

Bibliographic databases:

UDC: 519.214+519.212.2

Citation: A. N. Timashev, “Local limit theorems for one class of distributions in probabilistic combinatorics”, Diskr. Mat., 29:2 (2017), 109–132; Discrete Math. Appl., 28:6 (2018), 405–420

Citation in format AMSBIB
\Bibitem{Tim17} \by A.~N.~Timashev \paper Local limit theorems for one class of distributions in probabilistic combinatorics \jour Diskr. Mat. \yr 2017 \vol 29 \issue 2 \pages 109--132 \mathnet{http://mi.mathnet.ru/dm1422} \crossref{https://doi.org/10.4213/dm1422} \elib{http://elibrary.ru/item.asp?id=29437299} \transl \jour Discrete Math. Appl. \yr 2018 \vol 28 \issue 6 \pages 405--420 \crossref{https://doi.org/10.1515/dma-2018-0036} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000452905000006} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85058798322}