Renewal sequences and record chains related to multiple zeta sums

Abstract

For the random interval partition of [ 0 , 1 ] [0,1] generated by the uniform stick-breaking scheme known as GEM ( 1 ) (1) , let u k u_k be the probability that the first k k intervals created by the stick-breaking scheme are also the first k k intervals to be discovered in a process of uniform random sampling of points from [ 0 , 1 ] [0,1] . Then u k u_k is a renewal sequence. We prove that u k u_k is a rational linear combination of the real numbers 1 , ζ ( 2 ) , … , ζ ( k ) 1, \zeta (2), \ldots , \zeta (k) where ζ \zeta is the Riemann zeta function, and show that u k u_k has the limit 1 / 3 1/3 as k → ∞ 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 ( θ ) (\theta ) model, with beta ( 1 , θ ) (1,\theta ) instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.