Exact Convergence Rate for the Maximum of Standardized Gaussian Increments

Abstract

We prove an almost sure limit theorem on the exact convergence rate of the maximum of standarized gaussian random walk increments. This gives a more precise version of Shao's theorem (Shao, Q.-M., 1995. On a conjecture of Revesz. Proc. Amer. Math. Soc. 123, 575-582) in the gaussian case.