The exponential non-uniform bound on the half-normal approximation for the number of returns to the origin

Abstract

This research explored the number of returns to the origin within the framework of a symmetric simple random walk. Our primary objective was to approximate the distribution of return events to the origin by utilizing the half-normal distribution, which is chosen for its appropriateness as a limit distribution for nonnegative values. Employing the Stein's method in conjunction with concentration inequalities, we derived an exponential non-uniform bound for the approximation error. This bound signifies a significant advancement in contrast to existing bounds, encompassing both the uniform bounds proposed by Döbler <sup>[<span class="xref"><a href="#b1" ref-type="bibr">1</a></span>]</sup> and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee <sup>[<span class="xref"><a href="#b2" ref-type="bibr">2</a></span>]</sup>, and Siripraparat and Neammanee <sup>[<span class="xref"><a href="#b3" ref-type="bibr">3</a></span>]</sup>.