Convergence of bi-measure R-tree and the pruning process

TL;DR

This paper introduces a unified framework linking discrete and continuous tree pruning processes through a new topology on bi-measure trees, providing analytical characterization and examples including finite variance cases.

Contribution

It develops a new topology on bi-measure trees to unify discrete and continuous pruning dynamics as a single Markov process with analytical and continuity properties.

Findings

Unified topology links discrete and continuous pruning processes.

Analytical characterization via Markovian generator.

Examples include finite variance offspring case.

Abstract

In [Aldous,Pitman,1998] a tree-valued Markov chain is derived by pruning off more and more subtrees along the edges of a Galton-Watson tree. More recently, in [Abraham,Delmas,2012], a continuous analogue of the tree-valued pruning dynamics is constructed along L\'evy trees. In the present paper, we provide a new topology which allows to link the discrete and the continuous dynamics by considering them as instances of the same strong Markov process with different initial conditions. We construct this pruning process on the space of so-called bi-measure trees, which are metric measure spaces with an additional pruning measure. The pruning measure is assumed to be finite on finite trees, but not necessarily locally finite. We also characterize the pruning process analytically via its Markovian generator and show that it is continuous in the initial bi-measure tree. A series of examples is given, which include the finite variance offspring case where the pruning measure is the length measure on the underlying tree.