A Ray-Knight representation of up-down Chinese restaurants

Rogers, D. A.; Winkel, Matthias

doi:10.48550/arxiv.2006.06334

Cited by 7 publications

(18 citation statements)

References 26 publications

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“…Indeed, Shiga's approach could be understood as spindles-without-scaffolding. As discussed in [20,41], this scaffolding can be interpreted as representing a self-similar genealogy among spindles.…”

Section: 3mentioning

confidence: 99%

A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

Forman,

Rizzolo,

Shi

et al. 2020

Preprint

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We give a pathwise construction of a two-parameter family of purely-atomic-measure-valued diffusions in which ranked masses of atoms are stationary with the Poisson-Dirichlet(α, θ) distributions, for α ∈ (0, 1) and θ ≥ 0. This resolves a conjecture of Feng and Sun (2010). We build on our previous work on (α, 0)-and (α, α)interval partition evolutions. Indeed, we first extract a self-similar superprocess from the levels of stable processes whose jumps are decorated with squared Bessel excursions and distinct allelic types. We complete our construction by time-change and normalisation to unit mass. In a companion paper, we show that the ranked masses of the measure-valued processes evolve according to a two-parameter family of diffusions introduced by Petrov ( 2009), extending work of Ethier and Kurtz (1981). These ranked-mass diffusions arise as continuum limits of up-down Markov chains on Chinese restaurant processes.

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Section: 3mentioning

confidence: 99%

A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

Forman,

Rizzolo,

Shi

et al. 2020

Preprint

Self Cite

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“…The construction and analysis of ordered analogues of Petrov's [15] two-parameter extension of Ethier and Kurtz's [3] infinitely-many-neutral-alleles diffusion model has recently attracted significant interest in the literature [7,8,20,21,23]. Recall that the EKP(α, θ) diffusions constructed in [15] are a family of Feller diffusions on the closure of the Kingman simplex…”

Section: Introductionmentioning

confidence: 99%

“…Let (X (α,θ) n (k)) k≥0 be a Markov chain on C n with transition kernel T (α,θ) n defined as in Equation (1) using the p ↑ (α,θ) and p ↓ just described. A Poissonized version of this chain was considered in [21,23]. It can be shown that X (α,θ) n is an aperiodic, irreducible chain.…”

Section: Introductionmentioning

confidence: 99%

The leftmost column of ordered Chinese Restaurant Process up-down chains: intertwining and convergence

Rivera-Lopez¹,

Rizzolo²

2021

Preprint

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Recently there has been significant interest in constructing ordered analogues of Petrov's two-parameter extension of Ethier and Kurtz's infinitely-many-neutral-alleles diffusion model. One method for constructing these processes goes through taking an appropriate diffusive limit of Markov chains on integer compositions called ordered Chinese Restaurant Process up-down chains. The resulting processes are diffusions whose state space is the set of open subsets of the open unit interval. In this paper we begin to study nontrivial aspects of the order structure of these diffusions. In particular, for a certain choice of parameters, we take the diffusive limit of the size of the first component of ordered Chinese Restaurant Process up-down chains and describe the generator of the limiting process. We then relate this to the size of the leftmost maximal open subset of the open-set valued diffusions. This is challenging because the function taking an open set to the size of its leftmost maximal open subset is discontinuous. Our methods are based on establishing intertwining relations between the processes we study.

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“…We will address the remaining cases in a sequel paper [39], where we take a third approach to the threeparameter family. This involves the development of a limit theory for composition-valued Markov chains, which is of independent interest, and which in the two-parameter special case resolves a conjecture of [35]. See also [34] for a different approach to convergence results in the corresponding de-Poissonized setting, enhancing [27].…”

Section: Introductionmentioning

confidence: 99%

Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions

Shi¹,

Winkel²

2020

Preprint

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constructed (α, θ)-interval partition evolutions for α ∈ (0, 1) and θ ≥ 0, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension 2θ, where θ ≥ 0 acts as an immigration parameter. These evolutions have pseudo-stationary distributions related to regenerative Poisson-Dirichlet interval partitions. In this paper we study symmetry properties of (α, θ)-interval partition evolutions. Furthermore, we introduce a three-parameter family SSIP (α) (θ1, θ2) of self-similar interval partition evolutions that have separate left and right immigration parameters θ1 ≥ 0 and θ2 ≥ 0. They also have squared Bessel total mass processes of dimension 2θ, where θ = θ1 + θ2 − α ≥ −α covers emigration as well as immigration. Under the constraint max{θ1, θ2} ≥ α, we prove that an SSIP (α) (θ1, θ2)-evolution is pseudo-stationary for a new distribution on interval partitions, whose ranked sequence of lengths has Poisson-Dirichlet distribution with parameters α and θ, but we are unable to cover all parameters without developing a limit theory for composition-valued Markov chains, which we do in a sequel paper.

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A Ray-Knight representation of up-down Chinese restaurants

Cited by 7 publications

References 26 publications

A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

A two-parameter family of measure-valued diffusions with Poisson-Dirichlet stationary distributions

The leftmost column of ordered Chinese Restaurant Process up-down chains: intertwining and convergence

Two-sided immigration, emigration and symmetry properties of self-similar interval partition evolutions

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