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 Probl. Peredachi Inf., 2010, Volume 46, Issue 2, Pages 66–90 (Mi ppi2016)

Large Systems

Large deviations for distributions of sums of random variables: Markov chain method

V. R. Fatalov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $\{\xi_k\}_{k=0}^\infty$ be a sequence of i.i.d. real-valued random variables, and let $g(x)$ be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $\mathbf P\{\frac1n\sum_{k=0}^{n-1}g(\xi_k)<d\}$, $n\to\infty$, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with $g(x)=x^p$, $p>0$, and exponential random variables with $g(x)=x$ for $x\ge0$.

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English version:
Problems of Information Transmission, 2010, 46:2, 160–183

Bibliographic databases:

UDC: 621.391.1+519.2
Revised: 11.12.2009

Citation: V. R. Fatalov, “Large deviations for distributions of sums of random variables: Markov chain method”, Probl. Peredachi Inf., 46:2 (2010), 66–90; Problems Inform. Transmission, 46:2 (2010), 160–183

Citation in format AMSBIB
\Bibitem{Fat10} \by V.~R.~Fatalov \paper Large deviations for distributions of sums of random variables: Markov chain method \jour Probl. Peredachi Inf. \yr 2010 \vol 46 \issue 2 \pages 66--90 \mathnet{http://mi.mathnet.ru/ppi2016} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2724797} \elib{http://elibrary.ru/item.asp?id=15336968} \transl \jour Problems Inform. Transmission \yr 2010 \vol 46 \issue 2 \pages 160--183 \crossref{https://doi.org/10.1134/S0032946010020055} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000280241600005} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77956162588} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
2. V. R. Fatalov, “Ergodic means for large values of $T$ and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izv. Math., 77:6 (2013), 1224–1259
3. Kasparaviciute A., Deltuviene D., “Asymptotic Expansion For the Distribution Density Function of the Compound Poisson Process in Large Deviations”, J. Theor. Probab., 30:4 (2017), 1655–1676
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