Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function

TL;DR

This paper develops new non-asymptotic lower bounds for the upper tail probabilities of Gaussian process suprema, with applications to random multiplicative functions and classical constants in extreme value theory.

Contribution

It introduces novel lower bound techniques for Gaussian process tails that improve understanding of related probabilistic and number-theoretic problems.

Findings

Stronger lower bounds for Gaussian supremum tails.

Improved bounds for sums of random multiplicative functions.

Enhanced estimates for Pickands constants as alpha approaches zero.

Abstract

We prove new lower bounds for the upper tail probabilities of suprema of Gaussian processes. Unlike many existing bounds, our results are not asymptotic, but supply strong information when one is only a little into the upper tail. We present an extended application to a Gaussian version of a random process studied by Halasz. This leads to much improved lower bound results for the sum of a random multiplicative function. We further illustrate our methods by improving lower bounds for some classical constants from extreme value theory, the Pickands constants $H_{\alpha}$, as $\alpha\rightarrow 0$.