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This article is cited in 5 scientific papers (total in 5 papers)
Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method
V. R. Fatalov M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
We prove results on exact asymptotics as $n\to\infty$ for the expectations $\mathsf{E}_a \exp\bigl\{-\theta\sum_{k=0}^{n-1} g(X_k)\bigr\}$ and probabilities $\mathsf{P}_a\bigl\{\frac{1}{n}\sum_{k=0}^{n-1}g(X_k)<d\bigr\}$, where $\{\xi_k\}_{k=1}^\infty $ is a sequence of independent identically Laplace-distributed random variables, $X_n=X_0+\sum_{k=1}^n \xi_k$, $n\geqslant 1$, is the corresponding random walk on $\mathbb{R}$, $g(x)$ is a positive continuous function satisfying certain conditions, and $d>0$, $\theta>0$, $a\in\mathbb{R}$ are fixed numbers. Our results are obtained using a new method which is developed in this paper: the Laplace method for the occupation time of discrete-time Markov chains. For $g(x)$ one can take $|x|^p$, $\log(|x|^p+1)$, $p>0$, $|x|\log(|x|+1)$, or $e^{\alpha |x|}-1$, $0<\alpha<1/2$, $x\in\mathbb{R}$, for example. We give a detailed treatment of the case when $g(x)=|x|$ using Bessel functions to make explicit calculations.
Keywords:
large deviations, Markov chains, Laplace method, action functional,
occupation time, Bessel function.
Received: 25.11.2008
Citation:
V. R. Fatalov, “Exact asymptotics of probabilities of large deviations for Markov chains: the Laplace method”, Izv. Math., 75:4 (2011), 837–868
Linking options:
https://www.mathnet.ru/eng/im4061https://doi.org/10.1070/IM2011v075n04ABEH002554 https://www.mathnet.ru/eng/im/v75/i4/p189
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Abstract page: | 628 | Russian version PDF: | 187 | English version PDF: | 24 | References: | 82 | First page: | 5 |
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