TY - JOUR

T1 - Random Dirichlet series arising from records

AU - Peled, Ron

AU - Peres, Yuval

AU - Pitman, Jim

AU - Tanaka, Ryokichi

N1 - Publisher Copyright:
© 2015 The Mathematical Society of Japan.

PY - 2015

Y1 - 2015

N2 - We study the distributions of the random Dirichlet series with parameters (s; β) defined by [equation presented] where (In) is a sequence of independent Bernoulli random variables, In taking value 1 with probability 1=nβ 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 < β ≤ 1 with s + β > 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 β = 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 < β < 1 with s + β > 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.

AB - We study the distributions of the random Dirichlet series with parameters (s; β) defined by [equation presented] where (In) is a sequence of independent Bernoulli random variables, In taking value 1 with probability 1=nβ 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 < β ≤ 1 with s + β > 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 β = 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 < β < 1 with s + β > 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.

KW - Random Dirichlet series

KW - Records

KW - The van der Corput lemma

UR - http://www.scopus.com/inward/record.url?scp=84947087808&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84947087808&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06741705

DO - 10.2969/jmsj/06741705

M3 - Article

AN - SCOPUS:84947087808

VL - 67

SP - 1705

EP - 1723

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 4

ER -