Quantitative approximation of the invariant distribution of a Markov chain. A new approach
Résumé
In this paper, we deal with a Markov chain on a measurable state space $(\mathbb{X},\mathcal{X})$ which has a transition kernel $P$ admitting some small-set $S \in \mathcal{X}$, that is such that $P(x,A) \ge \nu(1_A) 1_S(x)$ for any $x \in \mathbb{X}$, $A \in \mathcal{X}$ and for some positive measure $\nu$. Under this condition, we propose a constructive characterisation of the existence of an $P$-invariant probability measure $\pi$ on $(\mathbb{X},\mathcal{X})$ such that $\pi(1_S)>0$. When such an $\pi$ exists, it is approximated in weighted or standard total variation norms by a finite linear combination of non-negative measures only depending on $\nu$, $P$ and $S$. Next, using standard drift-type conditions, we provide geometric/subgeometric convergence bounds of the approximation. Theses bounds are fully explicit and are as simple as possible. The rates of convergence are accurate, and they are optimal in the atomic case. Note that the rate of convergence for approximating the iterates of $P$ by the finite-rank submarkovian kernels introduced in [HerLedECP20] is also discussed. This is a new approach for approximating $\pi$ in the sense that it is not based on the convergence of the iterates of $P$ to $\pi$. Thus we need no aperiodicity condition. Moreover, the proofs are direct. They use neither the split chain in the non-atomic case, nor the renewal theory, nor the coupling method, nor the spectral theory. In some sense, this approach for Markov chains with a small-set is self-contained.
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