Rates of forgetting for the sequentially Markov coalescent

TL;DR

This paper analyzes the rate at which the sequentially Markov coalescent process loses memory of its initial state, providing theoretical bounds and implications for genetic linkage studies.

Contribution

It establishes geometric ergodicity for the embedded chain and decay rates for the continuous process, justifying common heuristic assumptions in population genetics.

Findings

Embedded jump chain is geometrically ergodic with explicit constants.

Total variation distance for the continuous process decays as 1/ell in genetic distance.

Results support heuristic approximations treating distant loci as independent.

Abstract

The sequentially Markov coalescent (SMC) is a Markov jump process which models correlations in local genealogies across a chromosome. It has been used as a theoretical tool for studying linkage disequilibrium and identity-by-descent, and it also forms the basis of a class of statistical procedures for estimating population history and inferring ancestry. In this paper, we study the rate at which SMC forgets its initial condition in the pairwise setting. For the embedded jump chain, we prove geometric ergodicity in total variation, with explicit constants. For the continuous process, by contrast, the total variation distance from stationarity decays as $\asymp 1/\ell$ in genetic distance $\ell$. We obtain analogous results for the closely related SMC' process using a novel time-change argument. One application of these results is to justify heuristic approximations used in the literature that treat distant loci as evolving independently.