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UID:8045@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20150618T110000
DTEND;TZID=Europe/Paris:20150618T120000
DTSTAMP:20241120T210006Z
URL:https://www.i2m.univ-amu.fr/evenements/random-dirichlet-series-arising
 -from-records-ryokichi-tanaka/
SUMMARY:Ryokichi Tanaka (Tohoku University\, Sendai): Random Dirichlet seri
 es arising from records - Ryokichi Tanaka
DESCRIPTION:Ryokichi Tanaka: We study the distributions of the random Diric
 hlet series with parameters $(s\, \\beta)$ defined by\n$$\nS=\\sum_{n=1}^{
 \\infty}\\frac{I_n}{n^s}\,\n$$\nwhere $(I_n)$ is a sequence of independent
  Bernoulli random variables\, $I_n$ taking value $1$ with probability $1/n
 ^\\beta$ and value $0$ otherwise.\nRandom series of this type are motivate
 d by the record indicator sequences which have been studied in extreme val
 ue theory in statistics.\nWe show that when $s&gt\;0$ and $0&lt\; \\beta \
 \le 1$ with $s+\\beta&gt\;1$ the distribution of $S$ has a density\; other
 wise it is purely atomic or not defined because of divergence.\nIn particu
 lar\, in the case when $s&gt\;0$ and $\\beta=1$\, we prove that for every 
 $0&lt\;s&lt\;1$ the density is bounded and continuous\, whereas for every 
 $s&gt\;1$ it is unbounded.\nIn the case when $s&gt\;0$ and $0&lt\;\\beta&l
 t\;1$ with $s+\\beta&gt\;1$\, the density is smooth.\nTo show the absolute
  continuity\, we obtain estimates of the Fourier transforms\, employing va
 n der Corput's method to deal with number-theoretic problems.\nWe also giv
 e 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.\n\nhttp://www.wpi-aimr.tohoku.ac.jp/mathematics_unit/ryoki
 chi_tanaka/index.html
ATTACH;FMTTYPE=image/jpeg:https://www.i2m.univ-amu.fr/wp-content/uploads/2
 020/01/Ryokichi_Tanaka.jpg
CATEGORIES:Séminaire,Probabilités
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DTSTART:20150329T030000
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