Random Dirichlet series arising from records

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Date(s) - 18/06/2015
11 h 00 min - 12 h 00 min

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We study the distributions of the random Dirichlet series with parameters $(s, \beta)$ defined by
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 $01$ 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 non atomic singular distribution which is induced by the series restricted to the primes.


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