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

• Published in: Methodology and computing in applied probability Vol. 22; no. 3; pp. 905 - 925
• Main Authors: Hervé, Loic, Ledoux, James
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2020; Springer Nature B.V; Springer Verlag
• Subjects: Autoregressive processes; Business and Management; Convergence; Discretization; Economics; Electrical Engineering; Ergodic processes; Invariants; Life Sciences; Markov analysis; Markov chains; Mathematics; Mathematics and Statistics; Probability; Probability distribution; Statistics; Markov chain; 60J05; Autoregressive models; Rate of convergence; 60J22; 60J22 Keywords : Markov chain; AMS subject classification : 60J05
• ISSN: 1387-5841; 1573-7713
• DOI: 10.1007/s11009-019-09746-0

Let ( X n ) n ∈ ℕ be a V -geometrically ergodic Markov chain on a measurable space X with invariant probability distribution π . In this paper, we propose a discretization scheme providing a computable sequence ( π ̂ k ) k ≥ 1 of probability measures which approximates π as k growths to infinity. The probability measure π ̂ k is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of ( π ̂ k ) k ≥ 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.