Quasi-limiting estimates for periodic absorbed Markov chains

TL;DR

This paper provides exponential estimates for the asymptotic behavior of periodic absorbed Markov chains, including convergence to quasi-stationary distributions and spectral gap bounds, enhancing understanding of their long-term dynamics.

Contribution

It introduces criteria for exponential convergence of conditional distributions in periodic Markov chains with absorption and characterizes quasi-stationary distributions.

Findings

Exponential convergence of conditional distributions to periodic limit measures

Characterization of when the limit sequence is constant (quasi-stationary)

Bounds on spectral gaps and ergodicity estimates for the Q-process

Abstract

We consider periodic Markov chains with absorption. Applying to iterates of this periodic Markov chain criteria for the exponential convergence of conditional distributions of aperiodic absorbed Markov chains, we obtain exponential estimates for the periodic asymptotic behavior of the semigroup of the Markov chain. This implies in particular the exponential convergence in total variation of the conditional distribution of the Markov chain given non-absorption to a periodic sequence of limit measures and we characterize the cases where this sequence is constant, which corresponds to the cases where the conditional distributions converge to a quasi-stationary distribution. We also characterize the first two eignevalues of the semigroup and give a bound for the spectral gap between these eigenvalues and the next ones. Finally, we give ergodicity estimates in total variation for the Markov chain conditioned to never be absorbed, often called Q-process, and quasi-ergodicity estimates for the original Markov chain.