Corrected diffusion approximation for random walks conditioned to stay positive

TL;DR

This paper investigates the accuracy of normal approximation for the probabilities of a random walk conditioned to stay positive, providing a Berry-Esseen-type inequality and extending previous results to more general initial conditions.

Contribution

It introduces a corrected diffusion approximation for random walks conditioned to stay positive, extending prior work and offering new bounds on approximation quality.

Findings

Derived a Berry-Esseen-type inequality for conditioned probabilities

Extended previous results to general initial positions

Complemented classical diffusion approximation results

Abstract

Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,\tau_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|\tau_x>n)$. Our Theorem 1 is an extension of the results in our previous paper (arXiv:2412.08502) where we have considered the special case $x=0$. It is also worth mentioning that Theorem 1 complements the results of Siegmund and Yuh (1982) on the corrected diffusion approximation.