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- Peled Ron
- Tel Aviv University, School of Mathematical Sciences
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- Peres Yuval
- Microsoft Research
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- Pitman Jim
- Department of Statistics, University of California Berkeley
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- Tanaka Ryokichi
- Tohoku University
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抄録
We study the distributions of the random Dirichlet series with parameters (s, β) defined by<img align="middle" src="./Graphics/abst-1705.jpg"/>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.
収録刊行物
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- Journal of the Mathematical Society of Japan
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Journal of the Mathematical Society of Japan 67 (4), 1705-1723, 2015
一般社団法人 日本数学会
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詳細情報 詳細情報について
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- CRID
- 1390001205115326976
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- NII論文ID
- 130005108957
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- NII書誌ID
- AA0070177X
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- ISSN
- 18811167
- 18812333
- 00255645
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- NDL書誌ID
- 026817736
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- 本文言語コード
- en
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- データソース種別
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- JaLC
- NDL
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