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) 1572915x (eISSN)
Vol. 12 1 1-17Subject Categories
Probability Theory and Statistics
DOI
10.1007/s10687-008-0065-3