Local probabilities for random walks with negative drift conditioned to stay nonnegative

Abstract

Let $S_n, n=0,1,...,$ with $S_0=0$ be a random walk with negative drift and let $\tau_x=\min\{k>0: S_k<-x\}, \, x\geq 0.$ Assuming that the distribution of i.i.d. increments of the random walk is absolutely continuous with subexponential density we find the asymptotic behavior, as $n\to\infty$ of the probabilities $\mathbf{P}(\tau_x=n)$ and $\mathbf{P}(S_n\in (y,y+\Delta],\tau_x>n)$ for fixed $x$ and various ranges of $y.$ The case of lattice distribution of increments is considered as well.