The elapsed time between two transient state observations for an absorbing Markov chain

TL;DR

This paper presents an exact method to compute the expected elapsed time and variance between two observations in an absorbing Markov chain, with applications in population genetics.

Contribution

It introduces a novel, exact solution for elapsed time calculation using the fundamental matrix of an absorbing Markov chain, avoiding diffusion approximation.

Findings

Provides a computable formula for expected elapsed time and variance

Applicable to population genetics for allele age estimation

Offers a new analytical approach for absorbing Markov chains

Abstract

Consider a system evolving according to an absorbing discrete-time Markov chain with known transition matrix. The state of the system is observed at two points in time, separated by an unknown number of generations. We are interested in calculating the expected elapsed time and its variance. We provide a novel, exact solution, which is computable from the fundamental matrix of a related absorbing Markov chain. This solution may be useful in population genetics for computing the expected age of a segregating allele without requiring diffusion approximation.