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October, 2015 Random Dirichlet series arising from records
Ron PELED, Yuval PERES, Jim PITMAN, Ryokichi TANAKA
J. Math. Soc. Japan 67(4): 1705-1723 (October, 2015). DOI: 10.2969/jmsj/06741705

Abstract

We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by

$S=\sum_{n=1}^{\infty}\frac{I_n}{n^s},$

where $(I_n)$ is a sequence of independent Bernoulli random variables, $I_n$ taking value 1 with probability $1/n^\beta$ and value 0 otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in extreme value theory in statistics. We show that when $s$ > 0 and 0 < $\beta \le 1$ with $s+\beta$ > 1 the distribution of $S$ has a density; otherwise it is purely atomic or not defined because of divergence. In particular, in the case when $s$ > 0 and $\beta=1$, we prove that for every 0 < $s$ < 1 the density is bounded and continuous, whereas for every $s$ > 1 it is unbounded. In the case when $s$ > 0 and 0 < $\beta$ < 1 with $s+\beta$ > 1, the density is smooth. To show the absolute continuity, we obtain estimates of the Fourier transforms, employing van der Corput's method to deal with number-theoretic problems. We also give further regularity results of the densities, and present an example of a non-atomic singular distribution which is induced by the series restricted to the primes.

Citation

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Ron PELED. Yuval PERES. Jim PITMAN. Ryokichi TANAKA. "Random Dirichlet series arising from records." J. Math. Soc. Japan 67 (4) 1705 - 1723, October, 2015. https://doi.org/10.2969/jmsj/06741705

Information

Published: October, 2015
First available in Project Euclid: 27 October 2015

zbMATH: 1346.37009
MathSciNet: MR3417510
Digital Object Identifier: 10.2969/jmsj/06741705

Subjects:
Primary: 37A45
Secondary: 42A38

Rights: Copyright © 2015 Mathematical Society of Japan

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Vol.67 • No. 4 • October, 2015
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