arXiv math/06115232006-11-17math.PR

On the equivalence of some eternal additive coalescents

Anne-Laure Basdevant

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

The eternal multiplicative coalescent encoding via excursions of Lévy-type processes, with Supplement

The multiplicative coalescent is a mean-field Markov process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version of the multiplicative coalescent was described in three different ways, one of which matched its (marginal) law to that of the ordered excursion lengths above past minima of a certain Lévy-type process. Using a modification of the breadth-first-walk construction from Aldous (1997) and Aldous and Limic (1998), and some new insight from the thesis by Uribe (2007), this work settles an open p…

2018-12-04math.PR

Eternal multiplicative coalescent is encoded by its Lévy-type processes

The multiplicative coalescent is a Markov process taking values in ordered $l^2$. It is a mean-field process in which any pair of blocks coalesces at rate proportional to the product of their masses. In Aldous and Limic (1998) each extreme eternal version $(\mathbf{X}(t),- \infty < t < \infty)$ of the multiplicative coalescent was described in three different ways. One of these specifications matches the (marginal) law of $\mathbf{X}(t)$ to that of the ordered excursion lengths above past minima of $\{L_{\mathbf{X}}(s) +ts, \,s \geq 0\}$, where $L_{\mathbf{X}}$ is a certain Lévy-type process w…

2016-10-17math.PR

On Moments of Multiplicative Coalescents

We prove existence of all moments of the multiplicative coalescent at all times. We obtain as byproducts a number of related results which could be of general interest. In particular, we show the finiteness of the second moment of the $l^2$ norm for any extremal eternal version of multiplicative coalescent. Our techniques are in part inspired by percolation, and in part are based on tools from stochastic analysis, notably the semi-martingale and the excursion theory.

2022-01-19math.PR

Some aspects of additive coalescents

We present some aspects of the so-called additive coalescence, with a focus on its connections with random trees, Brownian excursion, certain bridges with exchangeable increments, Lévy processes, and sticky particle systems.

2003-04-24math.PR

耦合 Bertoin 与 Aldous-Pitman 的加性凝聚表示

Coupling Bertoin's and Aldous-Pitman's representations of the additive coalescent

本文构造了加性凝聚的 Aldous-Pitman 与 Bertoin 两种表示的耦合，利用布朗 CRT 切割树的探索算法，统一了两种看似不同的构造。

2023-12-21math.PR

Asymptotics in Knuth's parking problem for caravans

We consider a generalized version of Knuth's parking problem, in which caravans consisting of a number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free space to park. Extending a recent work by Chassaing and Louchard, we relate the asymptotics for the sizes of blocks formed by occupied spots with the dynamics of the additive coalescent. According to the behavior of the caravan's size tail distribution, several qualitatively different versions of eternal additive coalescent are involved.

2005-02-10math.PR

更多相关论文收起更多论文

On a ternary coalescent process

We present a coalescent process where three particles merge at each coagulation step. Using a random walk representation, we prove duality with a fragmentation process, whose fragmentation law we specify explicitly. Furthermore, we give a second construction of the coalescence in terms of random binary forests and study asymptotic properties. Starting from N particles of unit mass, we obtain under an appropriate rescaling when N tends to infinity a well-known binary coalescence, the so-called standard additive coalescent.

2013-06-28math.PR

A new encoding of coalescent processes. Applications to the additive and multiplicative cases

We revisit the discrete additive and multiplicative coalescents, starting with $n$ particles with unit mass. These cases are known to be related to some "combinatorial coalescent processes": a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $(G(n,p))_p$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $n\to +\infty$. We propose to use the Prim order on the vertices instead of the…

2014-09-15cs.DMmath.COmath.PR

Scaling limits for the block counting process and the fixation line of a class of $Λ$-coalescents

We provide scaling limits for the block counting process and the fixation line of $Λ$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $Λ$ on $[0,1]$ satisfies $\int_{[0,1]}u^{-1}(Λ-bλ)({\rm d}u)<\infty$ for some $b>0$. Here $λ$ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The result is applied to beta coalescents with parameters $1$ and $b>0$. We split the generators into two parts…

2021-07-14math.PR

一般Dirichlet Xi-凝聚过程频率谱的渐近性质

Asymptotics of the frequency spectrum for general Dirichlet Xi-coalescents

本文研究一般Dirichlet Xi-凝聚过程的块计数过程短时渐近行为，证明其标度极限满足鞅问题且经Lamperti变换后为马尔可夫可加过程，并由此给出种群遗传学中n位点频谱的渐近结果。

2023-09-20math.PR

Bivariate Markov chains converging to Lamperti transform Markov Additive Processes

Motivated by various applications, we describe the scaling limits of bivariate Markov chains $(X,J)$ on $\mathbb Z_+ \times \{1,\ldots,κ\}$ where $X$ can be viewed as a position marginal and $\{1,\ldots,κ\}$ is a set of $κ$ types. The chain starts from an initial value $(n,i)\in \mathbb Z_+ \times \{1,\ldots,κ\}$, with $i$ fixed and $n \rightarrow \infty$, and typically we will assume that the macroscopic jumps of the marginal $X$ are rare, i.e. arrive with a probability proportional to a negative power of the current state. We also assume that $X$ is non-increasing. We then observe different …

2016-12-19math.PR