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 ${\mathcal{H}}_{\alpha }$. 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: ${\mathcal{H}}_1=1$ and ${\mathcal{H}}_2=1/\sqrt{\pi }$. 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 ${\mathcal{H}}_{\alpha }=1-(\alpha -1){\gamma }_{\mathrm{E}}+\mathcal{O}{\left(\alpha -1\right)}^2$.

13 pages, 3 figures

References in corpus (5)

• Extreme-Value Statistics of Fractional Brownian Motion Bridges
• The Maximum of a Fractional Brownian Motion: Analytic Results from Perturbation Theory
• Perturbative Expansion for the Maximum of Fractional Brownian Motion
• Probability distribution of the maximum of a smooth temporal signal
• Crossing intervals of non-Markovian Gaussian processes

Cited by in corpus (4)

• Extreme value statistics of correlated random variables: a pedagogical review
• First passage in an interval for fractional Brownian motion
• Long-term memory induced correction to Arrhenius law
• Kinetics of rare events for non-Markovian stationary processes and application to polymer dynamics