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Asymptotics of crossing probability of a boundary by the trajectory of a Markov chain. Exponentially decaying tails
A. A. Borovkov Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
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
$$
P\Big\{\max_{k\le n}\big(X(k)-g(k)\big)>0\Big\},
$$
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.
Keywords:
Markov chains, large deviations, boundary crossing, exponentially decaying tails, the law of iterated logarithm. Received: 17.12.2001
DOI:
10.4213/tvp284
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