Invariance principles for tree-valued Cannings chains

TL;DR

This paper proves the convergence of tree-valued Markov chains modeling genealogies in Cannings models to tree-valued Fleming-Viot processes, covering dust and dust-free cases with specific conditions.

Contribution

It establishes invariance principles for genealogical trees in Cannings models, extending convergence results to various types of Fleming-Viot processes with new assumptions.

Findings

Convergence holds for dust-free Fleming-Viot processes.

Convergence extends to processes with distance matrix distributions.

Additional assumptions ensure convergence to marked metric measure spaces.

Abstract

We consider sequences of tree-valued Markov chains that describe evolving genealogies in Cannings models, and we show their convergence in distribution to tree-valued Fleming-Viot processes. Under the conditions of M\"ohle and Sagitov, this convergence holds for all tree-valued Fleming-Viot processes under consideration in the dust-free case, and for the Fleming-Viot processes with values in the space of distance matrix distributions in the case with dust. Convergence to Fleming-Viot processes with values in the space of marked metric measure spaces in the case with dust is ensured by an additional assumption on the probability that a randomly sampled individual belongs to a non-singleton family.