Renewal sequences and record chains related to multiple zeta sums
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- by Jean-Jil Duchamps, Jim Pitman and Wenpin Tang PDF
- Trans. Amer. Math. Soc. 371 (2019), 5731-5755 Request permission
Abstract:
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 the limit $1/3$ as $k \rightarrow \infty$. 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.References
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Additional Information
- Jean-Jil Duchamps
- Affiliation: Collége de France, 11 place Marcelin Berthelot, 75231 Paris
- Email: jean-jil.duchamps@normalesup.org
- Jim Pitman
- Affiliation: Statistics Department, University of California, 367 Evans Hall, Berkeley, California 94720
- MR Author ID: 140080
- Email: pitman@stat.berkeley.edu
- Wenpin Tang
- Affiliation: Statistics Department, University of California, 367 Evans Hall, Berkeley, California 94720
- MR Author ID: 1108213
- Email: wenpintang@stat.berkeley.edu
- Received by editor(s): July 24, 2017
- Received by editor(s) in revised form: January 10, 2018
- Published electronically: September 18, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5731-5755
- MSC (2010): Primary 11M06, 60C05; Secondary 60E05
- DOI: https://doi.org/10.1090/tran/7516
- MathSciNet review: 3937308