Regenerative random permutations of integers

Pitman, Jim; Tang, Wenpin

doi:10.1214/18-aop1286

Cited by 34 publications

(48 citation statements)

References 96 publications

(138 reference statements)

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“…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.…”

Section: Introductionmentioning

confidence: 99%

On nested infinite occupancy scheme in random environment

Gnedin

Iksanov

2020

Probab. Theory Relat. Fields

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We consider an infinite balls-in-boxes occupancy scheme with boxes organised in nested hierarchy, and random probabilities of boxes defined in terms of iterated fragmentation of a unit mass. We obtain a multivariate functional limit theorem for the cumulative occupancy counts as the number of balls approaches infinity. In the case of fragmentation driven by a homogeneous residual allocation model our result generalises the functional central limit theorem for the block counts in Ewens' and more general regenerative partitions.

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Section: Introductionmentioning

confidence: 99%

On nested infinite occupancy scheme in random environment

Gnedin

Iksanov

2020

Probab. Theory Relat. Fields

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Let T be the first time that the configuration is a single wet component containing 1, and let K be its length. Pitman and Tang observed in [11] that the value of K describes the size of the first block in a family of regenerative permutations. Understanding block sizes has been useful for studying the structure of random Mallows permutations [1,6].…”

Section: Introductionmentioning

confidence: 99%

Poisson percolation on the square lattice

Cristali¹

2019

ALEA

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On the square lattice raindrops fall on an edge with midpoint x at rate x −α ∞ . The edge becomes open when the first drop falls on it. Let ρ(x, t) be the probability that the edge with midpoint x = (x 1 , x 2 ) is open at time t and let n(p, t) be the distance at which edges are open with probability p at time t. We show that with probability tending to 1 as t → ∞: (i) the cluster containing the origin C 0 (t) is contained in the square of radius n(p c − , t), and (ii) the cluster fills the square of radius n(p c + , t) with the density of points near x being close to θ(ρ(x, t)) where θ(p) is the percolation probability when bonds are open with probability p on Z 2 . Results of Nolin suggest that if N = n(p c , t) then the boundary fluctuations of C 0 (t) are of size N 4/7 .

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“…u k = P ∩ k i=1 (U j ∈ I i for some 1 ≤ j < n(k, 1)) (1.5) and we adopt the convention that u 0 := 1. This sequence (u k , k ≥ 0) is a renewal sequence appearing in the study of regenerative permutations in [51]. In that context it is easily shown that the limit u ∞ := lim k→∞ u k exists, but difficult to evaluate the u k for general k. However, computation of u k for the first few k = 1, 2, 3, .…”

Section: Introductionmentioning

confidence: 99%

Renewal sequences and record chains related to multiple zeta sums

Duchamps¹,

Pitman²,

Tang³

2018

Trans. Amer. Math. Soc.

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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.

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Regenerative random permutations of integers

Cited by 34 publications

References 96 publications

On nested infinite occupancy scheme in random environment

On nested infinite occupancy scheme in random environment

Poisson percolation on the square lattice

Renewal sequences and record chains related to multiple zeta sums

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