State-Discretization of V-Geometrically Ergodic Markov Chains and Convergence to the Stationary Distribution

Let ( X n ) n ∈ ℕ $(X_{n})_{n \in \mathbb {N}}$ be a V -geometrically ergodic Markov chain on a measurable space X $\mathbb {X}$ with invariant probability distribution π. In this paper, we propose a discretization scheme providing a computable sequence ( π ̂ k ) k ≥ 1 $(\widehat \pi _{k})_{k\ge 1}$ of probability measures which approximates π as k growths to infinity. The probability measure π ̂ k $\widehat \pi _{k}$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of ( π ̂ k ) k ≥ 1 $(\widehat \pi _{k})_{k\ge 1}$ to π is given. As a result, the specific case of first order autoregressive processes with linear and non-linear errors is studied. Finally, illustrations of the procedure for such autoregressive processes are provided, in particular when no explicit formula for π is known.