Random Dirichlet series arising from records

Carte non disponible
Speaker Home page :
Speaker :
Speaker Affiliation :

()

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

Catégories Pas de Catégories


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 $01$ 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.

[http://www.wpi-aimr.tohoku.ac.jp/mathematics_unit/ryokichi_tanaka/index.html]


Retour en haut 

Secured By miniOrange