Transition phenomena for ladder epochs of random walks with small negative drift

Details Transition phenomena for ladder epochs of random walks with small negative drift Transition phenomena for ladder epochs of random walks with small negative drift

2009-05-08

arxiv.org/abs/0905.1186v1

Mathematics

Probability

27 pages

Scientific paper

For a family of random walks $\{S^{(a)}\}$ satisfying
$\mathbf{E}S_1^{(a)}=-a<0$ we consider ladder epochs $\tau^{(a)}=\min\{k\geq1:
S_k^{(a)}<0\}$. We study the asymptotic, as $a\to0$, behaviour of
$\mathbf{P}(\tau^{(a)}>n)$ in the case when $n=n(a)\to\infty$. As a consequence
we obtain also the growth rates of the moments of $\tau^{(a)}$.

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