Berry-Esseen inequality for random walks conditioned to stay positive

TL;DR

This paper establishes a Berry-Esseen type estimate for the rate of convergence of zero-mean, finite-variance random walks conditioned to stay positive to the Rayleigh distribution, with a convergence rate of order n^{-1/2}.

Contribution

It provides the first explicit Berry-Esseen bound for conditioned random walks converging to the Rayleigh distribution.

Findings

Convergence rate of order n^{-1/2} for the conditioned walk to Rayleigh distribution.

Explicit Berry-Esseen estimate derived for the conditioned random walk.

Validation of the convergence rate under zero mean and finite variance assumptions.

Abstract

We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that they converge to the Rayleigh distribution. In the present paper we derive a Berry-Esseen type estimate and show that the rate of convergence is of order $n^{-1/2}$.