Pickands' constant $H_α$ does not equal $1/Γ(1/α)$, for small $α$

Pickands' constants $H_α$ appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps $H_α = 1/Γ(1/α)$ for all $0 < α\leq 2$, 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 $α$, and in fact that $H_α \geq (1.1527)^{1/α}/Γ(1/α)$ for all sufficiently small $α$. 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.

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