Precise large deviations of the first passage time

TL;DR

This paper investigates the precise asymptotic behavior of the probability that the sum of i.i.d. random variables first exceeds a linear boundary at a specific scaled time, refining classical large deviation estimates.

Contribution

It provides detailed large deviation results for the first passage time of partial sums crossing a linear boundary, extending classical ruin probability estimates.

Findings

Derived asymptotics for the probability of first crossing at a fixed scaled time.

Extended classical Cramér's estimate to precise large deviations.

Provided formulas for the asymptotic behavior of crossing probabilities.

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 $$ \tau_u = \inf\{n:\; S_n > u\}. $$ 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.