Renewal sequences and record chains related to multiple zeta sums

Duchamps, Jean-Jil; Pitman, Jim; Tang, Wenpin

doi:10.1090/tran/7516

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.

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