## Electronic Journal of Probability

### The Entrance Boundary of the Multiplicative Coalescent

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1454101763

Digital Object Identifier
doi:10.1214/EJP.v3-25

Mathematical Reviews number (MathSciNet)
MR1491528

Zentralblatt MATH identifier
0889.60080

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

Rights

#### Citation

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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