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

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 ^[1] and polynomial non-uniform bounds presented by Sama-ae, Chaidee, and Neammanee ^[2], and Siripraparat and Neammanee ^[3].