The Annals of Applied Probability

Practical drift conditions for subgeometric rates of convergence

Randal Douc, Gersende Fort, Eric Moulines, and Philippe Soulier

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We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a ψ-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224–247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings–Metropolis algorithms.

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Ann. Appl. Probab., Volume 14, Number 3 (2004), 1353-1377.

First available in Project Euclid: 13 July 2004

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Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Markov chains stationary distribution rate of convergence


Douc, Randal; Fort, Gersende; Moulines, Eric; Soulier, Philippe. Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab. 14 (2004), no. 3, 1353--1377. doi:10.1214/105051604000000323.

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  • AngoNze, P. (1994). Critères d'ergodicité de modèles markoviens. Estimation non-paramétrique sous des hypothèses de dépendance. Ph.D. dissertation, Univ. Paris 9, Dauphine.
  • AngoNze, P. (2000). Geometric and subgeometric rates for Markovian processes: A robust approach. Technical report, Univ. de Lille III.
  • Duflo, M. (1997). Random Iterative Systems. Springer, Berlin.
  • Fort, G. and Moulines, E. (2000). V-subgeometric ergodicity for a Hastings–Metropolis algorithm. Statist. Probab. Lett. 49 401–410.
  • Fort, G. and Moulines, E. (2003). Polynomial ergodicity of Markov transition kernels. Stochastic Process. Appl. 103 57–99.
  • Gourieroux, C. and Robert, C. (2001). Tails and extremal behaviour of stochastic unit root models. Technical report, Centre de Recherche en Economie et Statistique du Travail.
  • Granger, C. and Sawnson, N. (1997). An introduction to stochastic unit-root processes. J. Econometrics 80 35–62.
  • Grunwald, G., Hyndman, R., Tedesco, L. and Tweedie, R. (2000). Non-Gaussian conditional linear AR(1) models. Aust. N. Z. J. Stat. 42 479–495.
  • Jarner, S. and Hansen, E. (2000). Geometric ergodicity of Metropolis algorithms. Stochastic Process. Appl. 85 341–361.
  • Jarner, S. and Roberts, G. (2002). Polynomial convergence rates of Markov chains. Ann. Appl. Probab. 12 224–247.
  • Klokov, S. and Veretennikov, A. (2002). Sub-exponential mixing rate for a class of Markov processes. Technical Report 1, School of Mathematics, Univ. Leeds.
  • Malyshkin, M. (2001). Subexponential estimates of the rate of convergence to the invariant measure for stochastic differential equations. Theory Probab. Appl. 45 466–479.
  • Mengersen, K. and Tweedie, R. (1996). Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 101–121.
  • Meyn, S. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Nummelin, E. and Tuominen, P. (1983). The rate of convergence in Orey's theorem for Harris recurrent Markov chains with applications to renewal theory. Stochastic Process. Appl. 15 295–311.
  • Roberts, G. and Tweedie, R. (1996). Geometric convergence and central limit theorem for multidimensional Hastings and Metropolis algorithms. Biometrika 83 95–110.
  • Tanikawa, A. (2001). Markov chains satisfying simple drift conditions for subgeometric ergodicity. Stoch. Model. 17 109–120.
  • Tuominen, P. and Tweedie, R. (1994). Subgeometric rates of convergence of $f$-ergodic Markov chains. Adv. in Appl. Probab. 26 775–798.
  • Veretennikov, A. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115–127.
  • Veretennikov, A. (2000). On polynomial mixing and convergence rate for stochastic differential and difference equations. Theory Probab. Appl. 44 361–374.