Precise large deviations for the first passage time of a random walk with negative drift

Abstract:

Let $S_n$ be partial sums of an i.i.d. sequence $\{X_i\}$. We assume that $\mathbb {E} X_1 <0$ and $\mathbb {P}[X_1>0]>0$. In this paper we study the first passage time \begin{equation*} \tau _u = \inf \{n:\; S_n > u\}. \end{equation*} The classical Cramér’s estimate of the ruin probability says that \begin{equation*} \mathbb {P}[\tau _u<\infty ] \sim C e^{-\alpha _0 u}\qquad \text {as } u\to \infty , \end{equation*} for some parameter $\alpha _0$. The aim of the paper is to describe precise large deviations of the first crossing by $S_n$ a linear boundary. More precisely for a fixed parameter $\rho$ we study asymptotic behavior of $\mathbb {P}\big [\tau _u = \lfloor u/\rho \rfloor \big ]$ as $u$ tends to infinity.