Renewal sequences and record chains related to multiple zeta sums

Abstract

For the random interval partition of [0, 1] generated by the uniform stickbreaking 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, ζ(2), . . . , ζ(k) where ζ is the Riemann zeta function, and show that u k has limit 1/3 as k → ∞. 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(θ) model, with beta(1, θ) instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.

On nested infinite occupancy scheme in random environment

“…where N (t) := #{r ∈ N : T r ≤ t}. Using the last limit relation and (16) we conclude that the second summand converges to 0 a.s., as t → ∞. The first summand converges to zero in probability, as t → ∞, by Markov's inequality in combination with…”

“…having utilized (48) for the second inequality. In view of (16), this entails that a classical sufficient condition for tightness (formula (12.51) on p. 95 in [10]) holds…”

“…Residual allocation models with other choices of parameters for the U i 's with different beta distributions are found in [29,37]. Much effort has been devoted to the occupancy scheme, known as the Bernoulli sieve, which is based on a homogeneous residual allocation model (1), that is, with independent and identically distributed (iid) factors U i having arbitrary distribution on (0, 1), see [2,16,22,27,28,34]. The homogeneous model has a multiplicative regenerative property, also inherited by the partition of the set of balls.…”

Gaps and interleaving of point processes in sampling from a residual allocation model

“…The equivalence of formula (1.8) for general k ≥ 0 with the Markov property of Q • expressed in Theorem 1.2 is also easily checked. Further study of the chain Q • in the GEM(0, θ) case leads to a remarkable connection to the Riemann zeta function [10]. Theorem 1.2 will be proved in Section 3, along with the following corollaries.…”

Poisson percolation on the square lattice

“…Our inspiration comes from the rainstick process. It was introduced by Pitman and Tang in [11] with followup work by Pitman, Tang and Duchamps [4]. In this discrete time process raindrops fall one after the other on the positive integers and sites become wet when landed on.…”