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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 $$ τ_u = \inf\{n:\; S_n > u\}. $$ The classical Cramér's estimate of the ruin probability says that $$ \mathbb{P}[τ_u<\infty] \sim C e^{-α_0 u}\quad \text{as } u\to \infty, $$ for some parameter $α_0$. 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 $ρ$ we study asymptotic behavior of $\mathbb{P}\left[τ_u = \lfloor u/ρ\rfloor \right]$ as $u$ tends to infinity.