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

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 $\alpha \to 0$.

Published in at http://dx.doi.org/10.1214/12-AAP847 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Cited by in corpus (7)

• On asymptotic constants in the theory of extremes for Gaussian processes
• Extremes of a class of nonhomogeneous Gaussian random fields
• Helson's problem for sums of a random multiplicative function
• A note on multiplicative functions resembling the Möbius function
• Partial sums of random multiplicative functions and extreme values of a model for the Riemann zeta function
• Partial sums of biased random multiplicative functions
• Pseudomoments of the Riemann zeta function