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Extremes of Shepp statistics for Gaussian random walk

Dmitrii Zholud (Institutionen för matematiska vetenskaper)
Extremes (1386-1999). Vol. 12 (2009), 1, p. 1-17.
[Artikel, refereegranskad vetenskaplig]

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.

Nyckelord: Gaussian random walk increments, Shepp statistics, High excursions, Extreme values, Large deviations, Moderate deviations, Asymptotic, behavior, Distribution tail, Gumbel law, Limit theorems, Weak theorems, large jumps, law



Denna post skapades 2010-07-16.
CPL Pubid: 123885

 

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