Electronic Journal of Probability

The Entrance Boundary of the Multiplicative Coalescent

David Aldous and Vlada Limic

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The multiplicative coalescent $X(t)$ is a $l^2$-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From random graph asymptotics it is known (Aldous (1997)) that there exists a standard version of this process starting with infinitesimally small clusters at time $- \infty$.

In this paper, stochastic calculus techniques are used to describe all versions $(X(t);- \infty < t < \infty)$ of the multiplicative coalescent. Roughly, an extreme version is specified by translation and scale parameters, and a vector $c \in l^3$ of relative sizes of large clusters at time $- \infty$. Such a version may be characterized in three ways: via its $t \to - \infty$ behavior, via a representation of the marginal distribution $X(t)$ in terms of excursion-lengths of a Lévy-type process, or via a weak limit of processes derived from the standard version via a "coloring" construction.

Article information

Electron. J. Probab., Volume 3 (1998), paper no. 3, 59 pp.

Accepted: 19 January 1998
First available in Project Euclid: 29 January 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J50: Boundary theory
Secondary: 60J75: Jump processes

Markov process entrance boundary excursion Lévy process random graph stochastic coalescent weak convergence

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Aldous, David; Limic, Vlada. The Entrance Boundary of the Multiplicative Coalescent. Electron. J. Probab. 3 (1998), paper no. 3, 59 pp. doi:10.1214/EJP.v3-25. https://projecteuclid.org/euclid.ejp/1454101763

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