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


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.


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


Received: 1 August 2014; Revised: 1 July 2015; Published: February 2017
First available in Project Euclid: 27 September 2016

zbMATH: 1359.60051
MathSciNet: MR3556785
Digital Object Identifier: 10.3150/15-BEJ757

Keywords: Pickands’ constants , Stationary Gaussian processes , suprema of processes

Rights: Copyright © 2017 Bernoulli Society for Mathematical Statistics and Probability

Vol.23 • No. 1 • February 2017
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