# SMILE

## Renewal Sequences and Record Chains Related to Multiple Zeta Sums

### 2017

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.

# Bibtex

@article{DPT17,
title = {Renewal Sequences and Record Chains Related to Multiple Zeta Sums},
journal = {Transactions of the American Mathematical Society},
pages = {to appear},
author = {Duchamps, Jean-Jil and Pitman, Jim and Tang, Wenpin},
month = jul,
year = {2017},
doi = {10.1090/tran/7516},
url = {https://arxiv.org/abs/1707.07776}
}