On moments of multiplicative coalescents (Q6616042)

In this paper, the authors study higher moments of the multiplicative coalescent which is a Markov process that describes the merging of blocks with finite mass where the the rate of coalescence is equal to the product of the masses.\N\NThe main result states that all moments of the multiplicative coalescent at all times exist and finite. More specifically, the following is proved.\N\NIf \(\textbf{X}:=(\textbf{X}(t), t\geq 0)\) is a multiplicative coalescent process started with \(\textbf{x} \in l^2_{\searrow}\), then for any \(n \in \mathbb{N}\) and \(t\geq 0\),\N\begin{align*}\N\mathbb{E}\| \textbf{X}(t)\|^n <\infty.\N\end{align*}\NHere, \(\textbf{X}(t)=(X_1(t),X_2(t),\ldots)\), where \(X_j(t)\) is the size of the \(j\)th largest component at time \(t\), and \(l^2_{\searrow}\) denotes the space of all infinite sequences \(\textbf{x}=(x_1,x_2,\dots)\) with non-increasing components and \(\sum_{ j } x_j^2<\infty\), equipped with \(l^2\)-norm \(\|\cdot \|\).\N\NThe authors also investigate generalizations of Aldous standard (eternal) multiplicative coalescent. In particular, they show that any extremal eternal multiplicative coalescent has a finite second moment at any given time. Some auxiliary results include upper bounds for the probabilities of inter-connections for inhomogenous random graphs.