The Annals of Statistics

Local stationarity and time-inhomogeneous Markov chains

Lionel Truquet

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A primary motivation of this contribution is to define new locally stationary Markov models for categorical or integer-valued data. For this initial purpose, we propose a new general approach for dealing with time-inhomogeneity that extends the local stationarity notion developed in the time series literature. We also introduce a probabilistic framework which is very flexible and allows us to consider a much larger class of Markov chain models on arbitrary state spaces, including most of the locally stationary autoregressive processes studied in the literature. We consider triangular arrays of time-inhomogeneous Markov chains, defined by some families of contracting and slowly-varying Markov kernels. The finite-dimensional distribution of such Markov chains can be approximated locally with the distribution of ergodic Markov chains and some mixing properties are also available for these triangular arrays. As a consequence of our results, some classical geometrically ergodic homogeneous Markov chain models have a locally stationary version, which lays the theoretical foundations for new statistical modeling. Statistical inference of finite-state Markov chains can be based on kernel smoothing and we provide a complete and fast implementation of such models, directly usable by the practitioners. We also illustrate the theory on a real data set. A central limit theorem for Markov chains on more general state spaces is also provided and illustrated with the statistical inference in INAR models, Poisson ARCH models and binary time series models. Additional examples such as locally stationary regime-switching or SETAR models are also discussed.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 2023-2050.

Received: October 2017
Revised: June 2018
First available in Project Euclid: 21 May 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M05: Markov processes: estimation 62G08: Nonparametric regression
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Markov chains local stationarity


Truquet, Lionel. Local stationarity and time-inhomogeneous Markov chains. Ann. Statist. 47 (2019), no. 4, 2023--2050. doi:10.1214/18-AOS1739.

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  • [1] Al-Osh, M. A. and Alzaid, A. A. (1987). First-order integer-valued autoregressive (INAR($1$)) process. J. Time Series Anal. 8 261–275.
  • [2] Athreya, K. B. and Fuh, C. D. (1992). Bootstrapping Markov chains: Countable case. J. Statist. Plann. Inference 33 311–331.
  • [3] Avery, P. J. and Henderson, D. A. (1999). Detecting a changed segment in DNA sequences. J. R. Stat. Soc. Ser. C. Appl. Stat. 48 489–503.
  • [4] Bartoli, N. and Del Moral, P. (2001). Simulation et algorithmes stochastiques: une introduction avec applications. Editions Cépaduès.
  • [5] Billingsley, P. (1961). Statistical Inference for Markov Processes. Statistical Research Monographs. Vol. II. Univ. Chicago Press, Chicago, IL.
  • [6] Birr, S., Volgushev, S., Kley, T., Dette, H. and Hallin, M. (2017). Quantile spectral analysis for locally stationary time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 79 1619–1643.
  • [7] Bolch, G., Greiner, S., de Meer, H. and Trivedi, K. S. (2006). Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, 2nd ed. Wiley, Hoboken, NJ.
  • [8] Bradley, R. C. (2005). Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surv. 2 107–144.
  • [9] Bradley, R. C. (2007). Introduction to Strong Mixing Conditions. Vol. 2. Kendrick Press, Heber City, UT.
  • [10] Brillinger, D. R., Morettin, P. A., Irizarry, R. A. and Chiann, C. (2000). Some wavelet-based analyses of Markov chain data. Signal Process. 80 1607–1627.
  • [11] Cao, X.-R. (1998). The Maclaurin series for performance functions of Markov chains. Adv. in Appl. Probab. 30 676–692.
  • [12] Dahlhaus, R. (1997). Fitting time series models to nonstationary processes. Ann. Statist. 25 1–37.
  • [13] Dahlhaus, R., Richter, S. and Wu, W. B. (2000). A likelihood approximation for locally stationary processes. Ann. Statist. 28 1762–1794.
  • [14] Dahlhaus, R. and Subba Rao, S. (2006). Statistical inference for time-varying ARCH processes. Ann. Statist. 34 1075–1114.
  • [15] Davis, R. A., Holan, S. H., Lund, R. and Ravishanker, N., eds. (2016) Handbook of Discrete-Valued Time Series. Chapman & Hall/CRC Handbooks of Modern Statistical Methods. CRC Press, Boca Raton, FL.
  • [16] Dedecker, J. and Prieur, C. (2004). Coupling for $\tau$-dependent sequences and applications. J. Theoret. Probab. 17 861–885.
  • [17] Dobrushin, R. L. (1956). Central limit theorem for nonstationary Markov chains. II. Teor. Veroyatn. Primen. 1 365–425.
  • [18] Dobrushin, R. L. (1970). Definition of a system of random variables by means of conditional distributions. Teor. Veroyatn. Primen. 15 469–497.
  • [19] Douc, R., Moulines, E. and Rosenthal, J. S. (2004). Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab. 14 1643–1665.
  • [20] Douc, R., Moulines, E. and Stoffer, D. S. (2014). Nonlinear Time Series: Theory, Methods, and Applications with R Examples. Chapman & Hall/CRC Texts in Statistical Science Series. Chapman & Hall/CRC, Boca Raton, FL.
  • [21] Doukhan, P. (1994). Mixing. Properties and Examples. Lecture Notes in Statistics 85. Springer, New York.
  • [22] Ferré, D., Hervé, Y. and Ledoux, J. (2013). Regular perturbation of $V$-geometrically ergodic markov chains. J. Appl. Probab. 50 184–194.
  • [23] Du, J. G. and Li, Y. (1991). The integer-valued autoregressive $(\operatorname{INAR}(p))$ model. J. Time Series Anal. 12 129–142.
  • [24] Fokianos, K. and Moysiadis, T. (2017). Binary time series models driven by a latent process. Econ. Stat. 2 117–130.
  • [25] Fokianos, K., Rahbek, A. and Tjøstheim, D. (2009). Poisson autoregression. J. Amer. Statist. Assoc. 104 1430–1439.
  • [26] Fryzlewicz, P., Sapatinas, T. and Subba Rao, S. (2008). Normalized least-squares estimation in time-varying ARCH models. Ann. Statist. 36 742–786.
  • [27] Fryzlewicz, P. and Subba Rao, S. (2011). Mixing properties of ARCH and time-varying ARCH processes. Bernoulli 17 320–346.
  • [28] Granger, C. and Stărică, C. (2005). Nonstationarities in stock returns. Rev. Econ. Stat. 87 503–522.
  • [29] Hairer, M. and Mattingly, J. C. (2011). Yet another look at Harris’ ergodic theorem for Markov chains. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Progress in Probability 63 109–117. Birkhäuser/Springer, Basel.
  • [30] Hall, P. and Bura, E. (2004). Nonparametric methods of inference for finite-state, inhomogeneous Markov processes. Bernoulli 10 919–938.
  • [31] Hamilton, J. D. (1989). A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 357–384.
  • [32] Horváth, L. and Kokoszka, P. (2012). Inference for Functional Data with Applications. Springer Series in Statistics. Springer, New York.
  • [33] Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen.
  • [34] Kulperger, R. J. and Prakasa Rao, B. L. S. (1989). Bootstrapping a finite state Markov chain. Sankhyā Ser. A 51 178–191.
  • [35] Li, G., Guan, B., Li, W. K. and Yu, P. L. H. (2015). Hysteretic autoregressive time series models. Biometrika 102 717–723.
  • [36] Meyn, S. and Tweedie, R. L. (2009). Markov Chains and Stochastic Stability, 2nd ed. Cambridge Univ. Press, Cambridge.
  • [37] Moysiadis, T. and Fokianos, K. (2014). On binary and categorical time series models with feedback. J. Multivariate Anal. 131 209–228.
  • [38] Rajagopalan, B., Lall, U. and Tarboton, D. G. (1996). Nonhomogeneous Markov model for daily precipitation. J. Hydrol. Eng. 1 33–40.
  • [39] Richter, S. and Dahlhaus, R. (2017). Cross validation for locally stationary processes. Available at arXiv:1705.10046.
  • [40] Saloff-Coste, L. and Zúñiga, J. (2007). Convergence of some time inhomogeneous Markov chains via spectral techniques. Stochastic Process. Appl. 117 961–979.
  • [41] Saloff-Coste, L. and Zúñiga, J. (2011). Merging for inhomogeneous finite Markov chains, Part II: Nash and log-Sobolev inequalities. Ann. Probab. 39 1161–1203.
  • [42] Sarukkai, R. R. (2000). Link prediction and path analysis using Markov chains. Comput. Netw. 33 377–386.
  • [43] Seneta, E. (2006). Non-negative Matrices and Markov Chains. Springer Series in Statistics. Springer, New York.
  • [44] Stockis, J.-P., Franke, J. and Kamgaing, J. T. (2010). On geometric ergodicity of CHARME models. J. Time Series Anal. 31 141–152.
  • [45] Subba Rao, S. (2006). On some nonstationary, nonlinear random processes and their stationary approximations. Adv. in Appl. Probab. 38 1155–1172.
  • [46] Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford Statistical Science Series 6. The Clarendon Press, New York.
  • [47] Truquet, L. (2017). A perturbation analysis of some markov chains models with time-varying parameters. Available at arXiv:1610.09272.
  • [48] Truquet, L. (2019). Supplement to “Local stationarity and time-inhomogeneous Markov chains.” DOI:10.1214/18-AOS1739SUPP.
  • [49] Vergne, N. (2008). Drifting Markov models with polynomial drift and applications to DNA sequences. Stat. Appl. Genet. Mol. Biol. 7 Art. 6, 45.
  • [50] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [51] Vogt, M. (2012). Nonparametric regression for locally stationary time series. Ann. Statist. 40 2601–2633.
  • [52] Winkler, G. (1995). Image Analysis, Random Fields and Dynamic Monte Carlo Methods: A Mathematical Introduction. Applications of Mathematics (New York) 27. Springer, Berlin.
  • [53] Zhang, T. and Wu, W. B. (2015). Time-varying nonlinear regression models: Nonparametric estimation and model selection. Ann. Statist. 43 741–768.

Supplemental materials

  • Supplement to “Local stationarity and time-inhomogeneous Markov chains.”. Contains the proofs of all the results as well as a discussion of various Markov chains models satisfying the assumptions used in this paper.