分野別セミナー

Abstract:
I talk about the distributions of the random Dirichlet series with parameters (s, β) defined by

\[
S = \sum I_n/n^s,
\]
where (I_n) is an independent sequence of Bernoulli random variables taking value 1 with probability 1/n^β and 0 otherwise. Random series of this type are motivated by the record indicator sequences which have been studied in the extreme value theory in statistics. We show that the distributions have densities when s > 0 and 0 < β ≤ 1 with s+β > 1, and are purely atomic or not defined because of divergence otherwise. In particular, in the case when s > 0 and β = 1, we prove that the density is bounded and continuous when 0 < s < 1, and unbounded when s>1. In the case when s>0 and 0<β<1 with s+β>1, we prove that 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. I will also discuss about some related problems. No specific background is required (all the terminology will be explained).