Large Deviations of Shepp Statistics for Fractional Brownian Motion

Abstract

Define the incremental fractional Brownian field $B_H(s+\tau )-B_H(s),H\in (0,1)$, where $B_H(s)$ is a standard fractional Brownian motion with Hurst index $H\in (0,1)$. In this paper we derive the exact asymptotic behaviour of the maximum ${\max }_{(\tau ,s)\in [0,1]\times [0,T]}(B_H(s+\tau )-B_H(s))$ for any $H\in (0,1/2)$ complimenting thus the result of Zholud (2008) for the Brownian motion.

8 pages

References in corpus (3)

• Maxima of asymptotically Gaussian random fields and moderate deviation approximations to boundary crossing probabilities of sums of random variables with multidimensional indices
• Extreme-Value Analysis of Standardized Gaussian Increments
• An extension of the Lévy characterization to fractional Brownian motion

Cited by in corpus (3)

• Extremes of a class of nonhomogeneous Gaussian random fields
• Limit laws on extremes of non-homogeneous Gaussian random fields
• Extremes of a type of locally stationary Gaussian random fields with applications to Shepp statistics