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

TL;DR

This paper disproves the conjecture that Pickands' constant equals 1/Γ(1/α) for all small α, providing new lower bounds and refining existing methods with Brownian motion calculations.

Contribution

It proves that Pickands' constant does not equal 1/Γ(1/α) for small α and establishes a new lower bound using refined comparison techniques.

Findings

H_pickands' constant is greater than (1.1527)^{1/α}/Γ(1/α) for small α

The conjecture that H_α equals 1/Γ(1/α) is false for small α

Refined conditioning and comparison approach improves lower bounds

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 $0 < \alpha \leq 2$, 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 $H_{\alpha} \geq (1.1527)^{1/\alpha}/\Gamma(1/\alpha)$ 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.