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 does not 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.
Citation
Adam J. Harper. "Pickands’ constant $H_{\alpha}$ does not equal $1/\Gamma(1/\alpha)$, for small $\alpha$." Bernoulli 23 (1) 582 - 602, February 2017. https://doi.org/10.3150/15-BEJ757
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