Distributional approximations for the number of sign changes and the maximum value of a symmetric random walk

Two path statistics of a symmetric random walk, namely the number of sign changes and the maximum value, are approximated by a half normal distribution. This study establishes exponential non-uniform bounds for these statistics by integrating Stein’s method with concentration inequalities. In addition, certain analytical techniques are employed to derive sharp upper bounds for their moments. The obtained error bounds significantly improve upon the previous results, including uniform bounds and polynomial non-uniform bounds. Finally, applications with the graphical illustrations are presented to demonstrate the sharpness of the proposed bounds.