First-passage time asymptotics over moving boundaries for random walk bridges

TL;DR

This paper investigates the asymptotic behavior of the first-passage time over moving boundaries for a conditioned random walk with finite variance, revealing phase transitions influenced by boundary proximity.

Contribution

It provides a detailed analysis of how moving boundaries affect first-passage time asymptotics, including phase transition phenomena depending on boundary distance.

Findings

Asymptotic tail behavior follows a regularly varying function with exponent -1/2.

Moving boundary effects are captured by a slowly varying function.

Phase transition occurs near the return point, depending on boundary proximity.

Abstract

We study the asymptotic tail probability of the first-passage time over a moving boundary for a random walk conditioned to return to zero, where the increments of the random walk have finite variance. Typically, the asymptotic tail behavior may be described through a regularly varying function with exponent -1/2, where the impact of the boundary is captured by the slowly varying function. Yet, the moving boundary may have a stronger effect when the tail is considered at a time close to the return point of the random walk bridge. In the latter case, a phase transition appears in the asymptotics, of which the precise nature depends on the order of distance between zero and the moving boundary.