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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.)
##### Keywords:
space-homogeneous time-homogeneous Markov chains
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