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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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 $\alpha\rightarrow0$.

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

First available in Project Euclid: 12 February 2013

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Zentralblatt MATH identifier

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

Gaussian processes bounds on tail probabilities Pickands constants random multiplicative functions


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.

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