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

الأصل · EN

Pickands' constants Hα appear in various classical limit results about tail probabilities of suprema of Gaussian processes. It is an often quoted conjecture that perhaps Hα = 1/Γ(1/α) for all 0 < α≤ 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 Hα ≥ (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.