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.

Gumbel law

High excursions

large jumps

Gaussian random walk increments

Distribution tail

Extreme values

Shepp statistics

behavior

law

Limit theorems

Asymptotic

Weak theorems

Moderate deviations

Large deviations

Author

Dmitrii Zholud

Chalmers, Mathematical Sciences, Mathematical Statistics

University of Gothenburg

Extremes

1386-1999 (ISSN)

Vol. 12 1 1-17

Subject Categories

Probability Theory and Statistics

DOI

10.1007/s10687-008-0065-3

More information

Created

10/8/2017