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

Mathieu Delorme 1 Alberto Rosso 2 Kay Jörg Wiese 1

Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2017, 50, pp.16LT04

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 $\mathcal{H}_{\alpha}=1 - (\alpha-1) \gamma_{\rm E} + \mathcal{O}\!\left( \alpha-1\right)^{2}$.

  • 1. LPTENS - Laboratoire de Physique Théorique de l'ENS
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques