Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails

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
$$ P\Big\{\max_{k\le n}\big(X(k)-g(k)\big)>0\Big\}, $$
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
$$ \sup_{u\le 0}E e^{(u+\xi(u))\lambda}<\infty,\qquad \sup_{u\ge 0}E e^{\lambda\xi(u)}<\infty, $$
where $\xi(u)=X(u,1)-u$ is the increments of the chain at point $u$ in one step.
The present paper is a continuation of article [A. A. Borovkov, Theory Probab. Appl., 47 (2002), pp. 584–608], in which it is assumed that the tails of the distributions of $\xi(u)$ are regularly varying. Here we establish limit theorems describing under rather broad conditions the asymptotic behavior of the probabilities in question in the domains of large and normal deviations. Besides, asymptotic properties of the regeneration cycles to a positive atom are considered and an analog of the law of iterated logarithm is established.