## The Annals of Applied Probability

### 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 Halász. 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 $\alpha\rightarrow0$.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 2 (2013), 584-616.

Dates
First available in Project Euclid: 12 February 2013

https://projecteuclid.org/euclid.aoap/1360682023

Digital Object Identifier
doi:10.1214/12-AAP847

Mathematical Reviews number (MathSciNet)
MR3059269

Zentralblatt MATH identifier
1268.60075

#### Citation

Harper, Adam J. Bounds on the suprema of Gaussian processes, and omega results for the sum of a random multiplicative function. Ann. Appl. Probab. 23 (2013), no. 2, 584--616. doi:10.1214/12-AAP847. https://projecteuclid.org/euclid.aoap/1360682023

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