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

TL;DR

This paper derives a first-order expansion of Pickands' constant around Brownian motion, providing a new exact expression for the constant's behavior near the Hurst index of 1.

Contribution

It extends perturbative methods to include drift in fractional Brownian motion, yielding a novel first-order approximation of Pickands' constant near Brownian motion.

Findings

Derived the first-order expansion of Pickands' constant around Brownian motion.

Established the exact relation $ ext{H}_ ext{alpha} = 1 - ( ext{alpha}-1) ext{gamma}_ ext{E} + ext{O}(( ext{alpha}-1)^2)$.

Enhanced understanding of Gaussian process extremes near standard Brownian motion.

Abstract

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