Half-normal approximation for statistics of symmetric simple random walk

In 2013, Döbler used Stein’s method to obtain the uniform bounds in half-normal approximation for three statistics of a symmetric simple random walk; the maximum value, the number of returns to the origin and the number of sign changes up to a given time n. In this paper, we give the non-uniform bounds for these statistics by using Stein’s method and the concentration inequality approach.