Renewal sequences and record chains related to multiple zeta sums

TL;DR

This paper explores the connection between renewal sequences from GEM$(1)$ interval partitions and multiple zeta values, revealing rational combinations of zeta constants and probabilistic interpretations involving Markov chains.

Contribution

It establishes that renewal sequences in GEM$(1)$ are rational linear combinations of 1 and zeta values, and links these sequences to Markov chains with record-like structures, extending to GEM$( heta)$ models.

Findings

u_k is a rational combination of 1 and zeta(k) for GEM(1)

u_k converges to 1/3 as k increases

Probabilistic interpretations of multiple zeta values via Markov chains

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 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.