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

• Published in: AIMS mathematics Vol. 9; no. 7; pp. 19031 - 19048
• Main Authors: Siripraparat, Tatpon, Jongpreechaharn, Suporn
• Format: Journal Article
• Language: English
• Published: AIMS Press 01.01.2024
• Subjects: half-normal approximation; stein's method; symmetric simple random walk; the number of returns to the origin
• ISSN: 2473-6988; 2473-6988
• DOI: 10.3934/math.2024926

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 [ <a href="#b1" ref-type="bibr">1 ] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee [ <a href="#b2" ref-type="bibr">2 ] , and Siripraparat and Neammanee [ <a href="#b3" ref-type="bibr">3 ] .