Asymptotics of Crossing Probability of a Boundary by the Trajectory of a Markov Chain. Exponentially Decaying Tails

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 $${\bf P}\Big\{\max_{k\le n}\big(X(k)-g(k)\big) > 0\Big\},$$ 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 $$ \sup_{u\le 0}{\bf E} e^{(u+\xi(u))\lambda} < \infty,\qquad \sup_{u\ge 0}{\bf E} e^{\lambda\xi(u)}<\infty, $$ where $\xi(u)=X(u,1)-u$ is the increments of the chain at point u in one step. The present pap...