An extension of a convergence theorem for Markov chains arising in population genetics

Abstract

We analyse the 𝓁²(𝜋)-convergence rate of irreducible and aperiodic Markov chains with N-band transition probability matrix P and with invariant distribution 𝜋. This analysis is heavily based on two steps. First, the study of the essential spectral radius r_ess(P_{|𝓁²(𝜋)}) of P_{|𝓁²(𝜋)} derived from Hennion’s quasi-compactness criteria. Second, the connection between the spectral gap property (SG₂) of P on 𝓁²(𝜋) and the V-geometric ergodicity of P. Specifically, the (SG₂) is shown to hold under the condition α₀≔∑_m=−N^Nlim sup_i→+∞(P(i,i+m)P^*(i+m,i)^1∕2<1. Moreover, r_ess(P_{|𝓁²(𝜋)}≤α₀. Effective bounds on the convergence rate can be provided from a truncation procedure.