Pickands' constant $H_{\alpha }$ does not equal $1/\Gamma (1/\alpha )$, for small $\alpha$

Abstract

Pickands' constants $H_{\alpha }$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_{\alpha }=1/\Gamma (1/\alpha )$ for all $0<\alpha \le 2$, but it is also frequently observed that this doesn't seem compatible with evidence coming from simulations. We prove the conjecture is false for small $\alpha$, and in fact that $H_{\alpha }\ge (1.1527{)}^{1/\alpha }/\Gamma (1/\alpha )$ for all sufficiently small $\alpha$. The proof is a refinement of the "conditioning and comparison" approach to lower bounds for upper tail probabilities, developed in a previous paper of the author. Some calculations of hitting probabilities for Brownian motion are also involved.

19 pages