Precise large deviations of the first passage time

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 $$\tau_u = \inf\{n:\; S_n > u\}.$$ The classical Cram\'er's estimate of the ruin probability says that $$\mathbb{P}[\tau_u<\infty] \sim C e^{-\alpha_0 u}\quad \text{as } u\to \infty,$$ 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 $\mathbb{P}\left[\tau_u = \lfloor u/\rho\rfloor \right]$ as $u$ tends to infinity.