Renewal sequences and record chains related to multiple zeta sums
| Autor: | Jim Pitman, Wenpin Tang, Jean-Jil Duchamps |
|---|---|
| Přispěvatelé: | Laboratoire de Probabilités, Statistiques et Modélisations (LPSM (UMR_8001)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Department of Statistics [Berkeley], University of California [Berkeley], University of California-University of California |
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
General Mathematics
01 natural sciences Combinatorics 010104 statistics & probability symbols.namesake Chain (algebraic topology) FOS: Mathematics Mathematics - Combinatorics Partition (number theory) Number Theory (math.NT) 0101 mathematics Algebraic number Linear combination ComputingMilieux_MISCELLANEOUS Mathematics Real number Sequence Mathematics - Number Theory Markov chain Applied Mathematics Probability (math.PR) 010102 general mathematics 11M06 60C05 60E05 Riemann zeta function [MATH.MATH-PR]Mathematics [math]/Probability [math.PR] symbols Combinatorics (math.CO) Mathematics - Probability |
| Zdroj: | Transactions of the American Mathematical Society Transactions of the American Mathematical Society, American Mathematical Society, 2019, 371 (8), pp.5731-5755. ⟨10.1090/tran/7516⟩ |
| ISSN: | 0002-9947 |
| DOI: | 10.1090/tran/7516⟩ |
| Popis: | For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a rational linear combination of the real numbers $1, \zeta(2), \ldots, \zeta(k)$ where $\zeta$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$ as $k \to \infty$. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM$(\theta)$ model, with beta$(1,\theta)$ instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function. Comment: 25 pages. This paper is published by https://www.ams.org/journals/tran/2019-371-08/S0002-9947-2018-07516-X/ |
| Databáze: | OpenAIRE |
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