On Moments of Multiplicative Coalescents

TL;DR

This paper proves the existence of all moments for the multiplicative coalescent process at any time, using stochastic analysis techniques, and establishes finiteness of the second moment for extremal eternal versions.

Contribution

It introduces new methods combining percolation and stochastic analysis to analyze moments of the multiplicative coalescent, including extremal eternal versions.

Findings

All moments of the multiplicative coalescent exist at all times.

Finiteness of the second moment of the $l^2$ norm for extremal eternal versions.

Techniques combining percolation and stochastic analysis are effective for this analysis.

Abstract

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.