First-passage times for random walks with non-identically distributed increments

TL;DR

This paper studies the asymptotic behavior of first-passage times for non-identically distributed random walks, showing convergence to Brownian meander under certain conditions, extending classical results to more general settings.

Contribution

It extends the analysis of first-passage times to random walks with independent, non-identically distributed increments satisfying the Lindeberg condition, and establishes convergence to Brownian meander.

Findings

Asymptotic behavior of first-passage times over moving boundaries.

Convergence of rescaled random walks to Brownian meander.

Validation under Lindeberg condition for non-i.i.d. increments.

Abstract

We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.