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Large deviations for Markov chains in the positive quadrant. (English. Russian original) Zbl 1068.60034
Russ. Math. Surv. 56, No. 5, 803-916 (2001); translation from Usp. Mat. Nauk 56, No. 5, 3-116 (2001).
This paper deals with \(N\)-partially space-homogeneous time-homogeneous Markov chains \(X(y,n)\), \(n=0,1,2,\dots, X(y,0)=y,\) in the positive quadrant \(R^{2+}=\{x=(x_1,x_2):x_1\geq 0,x_2\geq 0\}.\) The main object of study is the asymptotic behaviour of the probabilities \(P(X(y,n)\in \delta(x))\) as \(n\to \infty\) and \(x=x(n)\to \infty,\) where \(\delta(x)=\{y\in R^2:x_i\leq y_i<x_i+\delta\), \(i=1,2\}.\) One of the main results says that \[ \ln P(X(y,n)\in \delta(x))\sim-sD(\gamma,\beta,T), \] where the derivation function \(D(\gamma,\beta,T)\) is written out in closed form, time \(n\) depends on \(s\) in such a way that there are limits \(\gamma,\beta,\) and \(T\) for which one has \(y\sim s\gamma,\;x\sim s\beta,\;n\sim sT.\) Results on the asymptotics of large deviation probabilities can be used in system reliability analysis to estimate overflow probabilities or the probability of reaching a high critical level, and as well in control theory to prove the existence of various exponential moments of stationary distributions.

MSC:
60F10 Large deviations
60J05 Discrete-time Markov processes on general state spaces
60G05 Foundations of stochastic processes
60K25 Queueing theory (aspects of probability theory)
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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