Limiting distribution of the continuity modulus for Gaussian processes with stationary increments

Zakhar Kabluchko

doi:10.1016/j.spl.2008.11.020

Abstract

Abstract Let { X ( t ) , t ∈ R } be a Gaussian process with stationary increments, zero mean and a.s. continuous paths, whose variogram γ ( t ) behaves like c | t | α , c > 0 , α ∈ ( 0 , 2 ) , as t → 0 . We show that the continuity modulus of X has asymptotically Gumbel distribution. In the case α = 2 , a non-Gumbel limiting distribution is obtained.

Highlights

Let {X(t), t ∈ R} be a stochastic process
We show that the continuity modulus of X has asymptotically Gumbel distribution
The continuity modulus {ωn, n > 0} of X on the interval [0, 1] is defined as ωn sup

Summary

Introduction

Let {X(t), t ∈ R} be a stochastic process. The continuity modulus {ωn, n > 0} of X on the interval [0, 1] is defined as ωn. Note that the above conditions are satisfied for the fractional Brownian motion having γ(t) = |t|α, as well as for the generalized Ornstein-Uhlenbeck process and the generalized Cauchy model, the latter two being stationary Gaussian processes having the covariance functions rOU(t) = e−|t|α, rCauchy(t) = (1 + |t|α)−β and variograms γOU(t) = 2(1 − e−|t|α), γCauchy(t) = 2(1 − (1 + |t|α)−β), where α ∈ (0, 2), β > 0. Another example is given by γ(t) = log(1 + |t|α), α ∈ (0, 2).

Extremes of Gaussian processes

Proofs of Theorem 1 and Theorem 2