Pickands' constant at first order in an expansion around Brownian motion

Abstract

In the theory of extreme values of Gaussian processes, many results are expressed in terms of the Pickands constant $\mathcal{H}_α$. This constant depends on the local self-similarity exponent $α$ of the process, i.e. locally it is a fractional Brownian motion (fBm) of Hurst index $H=α/2$. Despite its importance, only two values of the Pickands constant are known: ${\cal H}_1 =1$ and ${\cal H}_2=1/\sqrtπ$. Here, we extend the recent perturbative approach to fBm to include drift terms. This allows us to investigate the Pickands constant $\mathcal{H}_α$ around standard Brownian motion ($α=1$) and to derive the new exact result $\mathcal{H}_α=1 - (α-1) γ_{\rm E} + \mathcal{O}\!\left( α-1\right)^{2}$.

13 pages, 3 figures