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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 2, Pages 254–273 (Mi tvp284)

This article is cited in 1 scientific paper (total in 1 paper)

Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails

A. A. Borovkov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $X(n)=X(u,n)$, $n=0,1,\ldots$, be a time homogeneous ergodic real-valued Markov chain with transition probability $P(u,B)$ and initial value $u\equiv X(u,0)=X(0)$. We study the asymptotic behavior of the crossing probability of a given boundary $g(k)$, $k=0,1,\ldots,n$, by a trajectory $X(k)$, $k=0,1,\ldots,n$, that is the probability
$$P\{\max_{k\le n}(X(k)-g(k))>0\},$$
where the boundary $g(\cdot)$ depends, generally speaking, on $n$ and on a growing parameter $x$ in such a way that $\min_{k\le n}g(k)\to\infty$ as $x\to\infty$. The chain is assumed to be partially space-homogeneous, that is there exists $N\ge 0$ such that for $u>N$, $v>N$ the probability $P(u,dv)$ depends only on the difference $v-u$. In addition, it is assumed that there exists $\lambda>0$ such that
$$\sup_{u\le 0}E e^{(u+\xi(u))\lambda}<\infty,\qquad \sup_{u\ge 0}E e^{\lambda\xi(u)}<\infty,$$
where $\xi(u)=X(u,1)-u$ is the increments of the chain at point $u$ in one step.
The present paper is a continuation of article [A. A. Borovkov, Theory Probab. Appl., 47 (2002), pp. 584–608], in which it is assumed that the tails of the distributions of $\xi(u)$ are regularly varying. Here we establish limit theorems describing under rather broad conditions the asymptotic behavior of the probabilities in question in the domains of large and normal deviations. Besides, asymptotic properties of the regeneration cycles to a positive atom are considered and an analog of the law of iterated logarithm is established.

Keywords: Markov chains, large deviations, boundary crossing, exponentially decaying tails, the law of iterated logarithm.

DOI: https://doi.org/10.4213/tvp284

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English version:
Theory of Probability and its Applications, 2004, 48:2, 226–242

Bibliographic databases:

Received: 17.12.2001

Citation: A. A. Borovkov, “Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails”, Teor. Veroyatnost. i Primenen., 48:2 (2003), 254–273; Theory Probab. Appl., 48:2 (2004), 226–242

Citation in format AMSBIB
\Bibitem{Bor03} \by A.~A.~Borovkov \paper Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 2 \pages 254--273 \mathnet{http://mi.mathnet.ru/tvp284} \crossref{https://doi.org/10.4213/tvp284} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2015452} \zmath{https://zbmath.org/?q=an:1055.60069} \elib{http://elibrary.ru/item.asp?id=13449321} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 2 \pages 226--242 \crossref{https://doi.org/10.1137/S0040585X97980361} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000222357100003} 

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This publication is cited in the following articles:
1. F. G. Ragimov, F. D. Azizov, “The limit theorems for first passage time of Markov chain for nonlinear boundary”, Theory Probab. Appl., 57:1 (2013), 172–178
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