The Annals of Applied Probability

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

Adam J. Harper

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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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1360682023

Digital Object Identifier
doi:10.1214/12-AAP847

Mathematical Reviews number (MathSciNet)
MR3059269

Zentralblatt MATH identifier
1268.60075

Subjects
Primary: 60G15: Gaussian processes
Secondary: 60G70: Extreme value theory; extremal processes 11N64: Other results on the distribution of values or the characterization of arithmetic functions

Keywords
Gaussian processes bounds on tail probabilities Pickands constants random multiplicative functions

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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