Fragmentation of compositions and intervals

Bernoulli Sieve

Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer $n$. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stick-breaking interval partition of $[0,1]$. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts.

Coagulation Fragmentation Laws Induced By General Coagulations of Two-Parameter Poisson-Dirichlet Processes

Pitman~(1999) describes a duality relationship between fragmentation and coagulation operators. An explicit relationship is described for the two-parameter Poisson-Dirichlet laws, with parameters {\footnotesize $(α,θ)$} and $(β,θ/α)$, wherein $PD(α, θ)$ is coagulated by $PD(β,θ/α)$ for $0<α<1$, $0 \leqβ<1$ and $-β<θ/α$. This remarkable explicit agreement was obtained by combinatorial methods via exchangeable partition probability functions~(EPPF). This work discusses an alternative analysis which can feasibly extend the characterizations above to more general models of $PD(α,θ)$ coagulated wit…

On the equivalence of some eternal additive coalescents

In this paper, we study additive coalescents. Using their representation as fragmentation processes, we prove that the law of a large class of eternal additive coalescents is absolutely continuous with respect to the law of the standard additive coalescent on any bounded time interval.

源自条件稳定 Galton–Watson 树的碎裂过程的不变性原理

Invariance principle for fragmentation processes derived from conditioned stable Galton-Watson trees

本文研究条件临界稳定 Galton–Watson 树随机删边所得碎裂过程，证明其重标度后收敛到 $\alpha$-稳定 Lévy 树的比例切割碎裂，并通过稳定 Lévy 游程划分给出构造，推广 Brownian CRT 结果。

Diffusions on a space of interval partitions: The two-parameter model

We introduce and study interval partition diffusions with Poisson--Dirichlet$(α,θ)$ stationary distribution for parameters $α\in(0,1)$ and $θ\ge 0$. This extends previous work on the cases $(α,0)$ and $(α,α)$ and builds on our recent work on measure-valued diffusions. Our methods for dealing with general $θ\ge 0$ allow us to strengthen previous work on the special cases to include initial interval partitions with dust. In contrast to the measure-valued setting, we can show that this extended process is a Feller process improving on the Hunt property established in that setting. These processes…

无限可交换分区的凝结-碎裂双重性：来自耦合混合泊松物种采样模型的双重性

Coagulation-Fragmentation Duality of Infinitely Exchangeable Partitions from Coupled Mixed Poisson Species Sampling Models

本文提出了一种基于耦合混合泊松物种采样模型的新框架，通过四组件系统实现了无限可交换分区的凝结-碎裂双重性，解决了传统分区分析的难题。

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

Forman et al. (2020+) constructed $(α,θ)$-interval partition evolutions for $α\in(0,1)$ and $θ\ge 0$, in which the total sums of interval lengths ("total mass") evolve as squared Bessel processes of dimension $2θ$, where $θ\ge 0$ 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 ${\rm SSIP}^{(α)}(θ_1,θ_2)$ of self-similar interval partition evolutions that hav…

Regenerative Composition Structures

A new class of random composition structures (the ordered analog of Kingman's partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownia…