Verifiable Conditions for the Irreducibility and Aperiodicity of Markov Chains by Analyzing Underlying Deterministic Models

TL;DR

This paper establishes verifiable conditions for the irreducibility and aperiodicity of Markov chains modeled by nonlinear state space models, linking deterministic control properties to stochastic chain behavior under weaker assumptions.

Contribution

It introduces new criteria based on controllability rank conditions and the concept of steadily attracting states, extending previous results to models with discontinuities and weaker smoothness assumptions.

Findings

Conditions for irreducibility and aperiodicity are verified via controllability matrix rank.

Existence of a globally attracting state is equivalent to chain irreducibility.

Steadily attracting states are introduced as a new concept linking control and stochastic properties.

Abstract

We consider Markov chains that obey the following general non-linear state space model: $\Phi_{k+1} = F(\Phi_k, \alpha(\Phi_k, U_{k+1}))$ where the function $F$ is $C^1$ while $\alpha$ is typically discontinuous and $\{U_k: k \in \mathbb{Z}_{> 0} \}$ is an independent and identically distributed process. We assume that for all $x$, the random variable $\alpha(x, U_1)$ admits a density $p_x$ such that $(x, w) \mapsto p_x(w)$ is lower semi-continuous. We generalize and extend previous results that connect properties of the underlying deterministic control model to provide conditions for the chain to be $\varphi$-irreducible and aperiodic. By building on those results, we show that if a rank condition on the controllability matrix is satisfied for all $x$, there is equivalence between the existence of a globally attracting state for the control model and $\varphi$-irreducibility of the Markov chain. Additionally, under the same rank condition on the controllability matrix, we prove that there is equivalence between the existence of a steadily attracting state and the $\varphi$-irreducibility and aperiodicity of the chain. The notion of steadily attracting state is new. Those results hold under considerably weaker assumptions on the model than previous ones that would require $(x,u) \mapsto F(x,\alpha(x,u))$ to be $C^\infty$ (while it can be discontinuous here). Additionally the establishment of a necessary and sufficient condition for the $\varphi$-irreducibility and aperiodicity without a structural assumption on the control set is novel---even for Markov chains where $(x,u) \mapsto F(x,\alpha(x,u))$ is $C^\infty$. We illustrate that the conditions are easy to verify on a non-trivial and non-artificial example of Markov chain arising in the context of adaptive stochastic search algorithms to optimize continuous functions in a black-box scenario.