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Large-deviation probabilities for one-dimensional Markov chains. Part 1: Stationary distributions
A. A. Borovkov,
D. A. Korshunov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
In this paper, we consider time-homogeneous and asymptotically space-homogeneous Markov chains that take values on the real line and have an invariant measure. Such a measure always exists if the chain is ergodic. In this paper, we continue the study of the asymptotic properties of
$\pi([x,\infty))$ as
$x \rightarrow \infty$ for the invariant measure
$\pi$, which was started in [A. A. Borovkov, Stochastic Processes in Queueing Theory, Springer-Verlag, New York, 1976], [A. A. Borovkov, Ergodicity and Stability of Stochastic Processes, TVP Science Publishers, Moscow, to appear], and [A. A. Brovkov and D. Korshunov, Ergodicity in a sense of weak convergence, equilibrium-type identities and large deviations for Markov chains, in Probability Theory and Mathematical Statistics, Coronet Books, Philadelphia, 1984, pp. 89–98]. In those papers, we studied basically situations that lead to a purely exponential decrease of
$\pi([x,\infty))$. Now we consider two remaining alternative variants: the case of "power" decreasing of
$\pi([x,\infty))$ and the "mixed" case when
$\pi([x,\infty))$ is asymptotically
$l(x)e^{-\beta x}$, where
$l(x)$ is an integrable function regularly varying at infinity and
$\beta>0$.
Keywords:
Markov chain, invariant measure, rough and exact asymptoticbehavior of large-deviation probabilities. Received: 10.02.1995
DOI:
10.4213/tvp2769