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

Abstract

Let S n S_n be partial sums of an i.i.d. sequence { X i } \{X_i\} . We assume that E X 1 > 0 \mathbb {E} X_1 >0 and P [ X 1 > 0 ] > 0 \mathbb {P}[X_1>0]>0 . In this paper we study the first passage time τ u = inf { n : S n > u } . \begin{equation*} \tau _u = \inf \{n:\; S_n > u\}. \end{equation*} The classical Cramér’s estimate of the ruin probability says that P [ τ u > ∞ ] ∼ C e − α 0 u as  u → ∞ , \begin{equation*} \mathbb {P}[\tau _u>\infty ] \sim C e^{-\alpha _0 u}\qquad \text {as } u\to \infty , \end{equation*} for some parameter α 0 \alpha _0 . The aim of the paper is to describe precise large deviations of the first crossing by S n S_n a linear boundary. More precisely for a fixed parameter ρ \rho we study asymptotic behavior of P [ τ u = ⌊ u / ρ ⌋ ] \mathbb {P}\big [\tau _u = \lfloor u/\rho \rfloor \big ] as u u tends to infinity.