Rate of convergence of Nummelin-type representation of the invariant distribution of a Markov chain via the residual kernel

Abstract

Let P be a Markov kernel on a measurable state space $(\mathbb{X},\mathcal{X})$ admitting some small-set $S\in \mathcal{X}$, that is: $P(x,A)\ge \mathrm{\nu }(1_A)1_S(x)$ for any $x\in \mathbb{X}$, $A\in \mathcal{X}$ and for some positive measure ν. Let π be a P-invariant probability measure such that $\mathrm{\pi }(1_S)>0$. Using the non-negative residual kernel $R:=P-\mathrm{\nu }(\cdot )1_S$, we study the rate of convergence to π, in weighted or standard total variation norms, of normalized versions of the series ${\sum }_{n=1}^{+\mathrm{\infty }}\mathrm{\nu }\circ R^{n-1}$. Under drift-type conditions on R, we provide geometric/polynomial convergence bounds of the rate of convergence. Theses bounds are fully explicit and are as simple as possible. Their proofs do not require to introduce the split chain in the non-atomic case, the renewal theory, the coupling method, or the spectral theory.