## Journal of the Mathematical Society of Japan

### Random Dirichlet series arising from records

#### 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.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1705-1723.

Dates
First available in Project Euclid: 27 October 2015

https://projecteuclid.org/euclid.jmsj/1445951163

Digital Object Identifier
doi:10.2969/jmsj/06741705

Mathematical Reviews number (MathSciNet)
MR3417510

Zentralblatt MATH identifier
1346.37009

#### Citation

PELED, Ron; PERES, Yuval; PITMAN, Jim; TANAKA, Ryokichi. Random Dirichlet series arising from records. J. Math. Soc. Japan 67 (2015), no. 4, 1705--1723. doi:10.2969/jmsj/06741705. https://projecteuclid.org/euclid.jmsj/1445951163

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