Asymptotic behaviour of random tridiagonal Markov chains in biological applications

TL;DR

This paper analyzes the long-term behavior of random tridiagonal Markov chains in biological contexts, showing they have a unique random attractor that simplifies to a path, regardless of probabilistic or deterministic variations.

Contribution

It proves the existence of a singleton random attractor for these chains using the Hilbert metric, applicable to both stochastic and deterministic cases.

Findings

Existence of a singleton random attractor for the chains

The attractor is a random path or periodic path in deterministic cases

The proof is valid without relying on probabilistic properties

Abstract

Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.