Stochastic Models for the Inference of Life Evolution

Renewal Sequences and Record Chains Related to Multiple Zeta Sums

Duchamps, J., Pitman, J., Tang, W.

Transactions of the American Mathematical Society


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.


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 = {}

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