Corrected diffusion approximation for random walks conditioned to stay positive

TL;DR

This work addresses the accuracy of normal approximations for random walks conditioned to stay positive. It extends previous results from starting at zero to arbitrary nonnegative starting points, providing a Berry-Esseen type bound with a diffusion correction term that depends on the boundary crossing quantity $\mathbb{E}|x+S_{\tau_x}|$. The authors develop extensive preliminary estimates, introduce a smoothing technique, and employ a time reversal argument to prove the corrected diffusion bound, which sharpens understanding of the Rayleigh-type limit in the conditioning regime $x=o(\sqrt{n})$. They also show how the bound can be enhanced for lattice and absolutely continuous increments by invoking local CLTs, reducing the moment dependence from a cubic to a quadratic factor and yielding explicit improved rates. Overall, the results yield a practical, asymptotically accurate description of the random walk’s distribution under the positive conditioning, with direct connections to diffusion approximations and prior corrected-diffusion work.

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 $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,τ_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|τ_x>n)$. 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.

Theorems & Definitions (23)

• Theorem 1
• Corollary 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• proof