Extremes of Shepp statistics for Gaussian random walk

Abstract

Let (ξ i , i ≥ 1) be a sequence of independent standard normal random variables and let S k = k i=1 ξ i be the corresponding random walk. We studydetermine asymptotic expressions for P M (N) T > u when u, N and T → ∞ 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 (N) T when T, N → ∞ and present corresponding normalization sequences.

Large deviations of Shepp statistics for fractional Brownian motion

“…Various authors refer to the process {Z * H (τ, T ), τ ≥ 0} as the standardised Shepp statistics. Important results for Shepp statistics and related quantities can be found in Deheuvels and Devroye (1987), Siegmund and Venkatraman (1995), Dümbgen and Spokoiny (2001), Kabluchko and Munk (2008) and Zholud (2009).…”

Extremes of Shepp statistics for the Wiener process

“…One of the applications of the result derived in this paper is given in Zholud (2008). Let (ξ i , i ≥ 1) be standard normal random variables, and S k be the corresponding random walk,…”

“…It will be used when building limit theorems for Shepp statistics for a Gaussian random walk (Zholud 2008). As a corollary of the lemma stated in Section 3 we obtain the limiting Gumbel distribution for M T , when T → ∞.…”

Extreme-Value Analysis of Standardized Gaussian Increments

“…It is a consequence of a more general result due to Shao [36], who proved a conjecture of Révész [34, §14.3] (see also [38] for a simplification of Shao's proof and [24] for a related result). The next theorem may be viewed as a distributional convergence version of the Erdös-Renyi law of large numbers in the case of standard normal summands and is a consequence of a more general result of Komlós and Tusnády proved in [23] (see also [31,39,40]). We give a short proof of this theorem in Section 5.…”