## Bernoulli

- Bernoulli
- Volume 23, Number 1 (2017), 582-602.

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

#### 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.

#### Article information

**Source**

Bernoulli, Volume 23, Number 1 (2017), 582-602.

**Dates**

Received: August 2014

Revised: July 2015

First available in Project Euclid: 27 September 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1475001367

**Digital Object Identifier**

doi:10.3150/15-BEJ757

**Mathematical Reviews number (MathSciNet)**

MR3556785

**Zentralblatt MATH identifier**

1359.60051

**Keywords**

Pickands’ constants stationary Gaussian processes suprema of processes

#### Citation

Harper, Adam J. Pickands’ constant $H_{\alpha}$ does not equal $1/\Gamma(1/\alpha)$, for small $\alpha$. Bernoulli 23 (2017), no. 1, 582--602. doi:10.3150/15-BEJ757. https://projecteuclid.org/euclid.bj/1475001367