Rates of forgetting for the sequentially Markov coalescent
TLDR
This paper analyzes the forgetting rates of the sequentially Markov coalescent (SMC), showing how quickly genetic correlations decay across a chromosome.
Key contributions
- Proves geometric ergodicity for the SMC's embedded jump chain with explicit constants.
- Shows continuous SMC's total variation distance from stationarity decays as ~ 1/l in genetic distance.
- Obtains analogous forgetting rate results for the SMC' process using a novel time-change argument.
- Justifies heuristic approximations that treat distant genetic loci as evolving independently.
Why it matters
The SMC is crucial for understanding genetic correlations and inferring population history. This work provides a rigorous mathematical understanding of its memory properties. These findings validate common approximations in population genetics, improving the theoretical foundation of genetic analysis.
Original 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.
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