Extremes of Shepp statistics for Gaussian random walk
Journal article, 2009
Let (xi(i), i >= 1) be a sequence of independent standard normal random variables and let S-k = Sigma(k)(i=1)xi(i) be the corresponding random walk. We study the renormalized Shepp statistic M-T((N)) = 1/root N (1 <= k <= TN 1 <= L <= N)max max (Sk+L-1 - Sk-1) and determine asymptotic expressions for P(M-T((N)) > u) when u, N and T -> infinity in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of M-T((N)) when T, N -> infinity and present corresponding normalization sequences.
Gaussian random walk increments