Electronic Journal of Statistics

Stability and asymptotics for autoregressive processes

Likai Chen and Wei Biao Wu

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The paper studies infinite order autoregressive models for both temporal and spatial processes. We present sufficient conditions for the existence of stationary distributions. To understand the underlying dynamics and to capture the dependence structure, we introduce functional dependence measures and relate them with Lipschitz coefficients of the data-generating mechanisms. Our stability result allows both short- and long-range dependence. With functional dependence measures, we can establish an asymptotic theory for the underlying processes.

Article information

Electron. J. Statist., Volume 10, Number 2 (2016), 3723-3751.

Received: July 2016
First available in Project Euclid: 6 December 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62E20: Asymptotic distribution theory

Autoregressive models Markov process nonlinear time series stationarity functional dependence measure invariance principle


Chen, Likai; Wu, Wei Biao. Stability and asymptotics for autoregressive processes. Electron. J. Statist. 10 (2016), no. 2, 3723--3751. doi:10.1214/16-EJS1213. https://projecteuclid.org/euclid.ejs/1480993452

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  • [1] Aue, A., Berkes, I. and Horváth, L. (2006). Strong approximation for the sums of squares of augmented GARCH sequences., Bernoulli 12 583–608.
  • [2] Berkes, I., Liu, W. and Wu, W. B. (2014). Komlós–Major–Tusnády approximation under dependence., The Annals of Probability 42 794–817.
  • [3] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems., Journal of the Royal Statistical Society. Series B (Methodological) 192–236.
  • [4] Brunsdon, C., Fotheringham, A. S. and Charlton, M. (1998). Spatial nonstationarity and autoregressive models., Environment and Planning A 30 957–973.
  • [5] Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models., Econometric Theory 18 17–39.
  • [6] Csörgö, M. and Horváth, L. (1997)., Limit Theorems in Change-Point Analysis 18. John Wiley & Sons Inc.
  • [7] Diaconis, P. and Freedman, D. (1999). Iterated random functions., SIAM Review 41 45–76.
  • [8] Ding, Z., Granger, C. W. and Engle, R. F. (1993). A long memory property of stock market returns and a new model., Journal of Empirical Finance 1 83–106.
  • [9] Douc, R., Roueff, F. and Soulier, P. (2008). On the existence of some processes., Stochastic Processes and Their Applications 118 755–761.
  • [10] Duan, J.-C. (1997). Augmented GARCH (p, q) process and its diffusion limit., Journal of Econometrics 79 97–127.
  • [11] El Machkouri, M., Volný, D. and Wu, W. B. (2013). A central limit theorem for stationary random fields., Stochastic Processes and Their Applications 123 1–14.
  • [12] Fan, J. and Yao, Q. (2003)., Nonlinear Time Series: Nonparametric and Parametric Methods. Springer Science & Business Media.
  • [13] Giraitis, L., Kokoszka, P. and Leipus, R. (2000). Stationary ARCH models: dependence structure and central limit theorem., Econometric Theory 16 3–22.
  • [14] Giraitis, L., Leipus, R. and Surgailis, D. (2007). Recent advances in ARCH modelling. In, Long Memory in Economics 3–38. Springer.
  • [15] Giraitis, L., Leipus, R. and Surgailis, D. (2009). ARCH () models and long memory properties. In, Handbook of Financial Time Series 71–84. Springer.
  • [16] Granger, C. W. J. and Andersen, A. P. (1978)., An Introduction to Bilinear Time Series Models. Vandenhoeck & Ruprecht.
  • [17] Hörmann, S. (2008). Augmented GARCH sequences: dependence structure and asymptotics., Bernoulli 543–561.
  • [18] Jarner, S. and Tweedie, R. (2001). Locally contracting iterated functions and stability of Markov chains., Journal of Applied Probability 494–507.
  • [19] Jenish, N. and Prucha, I. R. (2009). Central limit theorems and uniform laws of large numbers for arrays of random fields., Journal of Econometrics 150 86–98.
  • [20] Jenish, N. and Prucha, I. R. (2012). On spatial processes and asymptotic inference under near-epoch dependence., Journal of Econometrics 170 178–190.
  • [21] Kazakevicius, V. and Leipus, R. (2002). On stationarity in the arch ([infty infinity]) model., Econometric Theory 18 1–16.
  • [22] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV’-s, and the sample DF. I., Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32 111–131.
  • [23] Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach., Econometrica: Journal of the Econometric Society 347–370.
  • [24] Paulik, M. J., Das, M. and Loh, N. (1992). Nonstationary autoregressive modeling of object contours., Signal Processing, IEEE Transactions on 40 660–675.
  • [25] Priestley, M. B. (1988)., Nonlinear and Nonstationary Time Series Analysis. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London.
  • [26] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics., Random Structures and Algorithms 9 223–252.
  • [27] Rao, T. S. and Gabr, M. (2012)., An Introduction to Bispectral Analysis and Bilinear Time Series Models 24. Springer Science & Business Media.
  • [28] Robinson, P. M. (1991). Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression., Journal of Econometrics 47 67–84.
  • [29] Shao, X. and Wu, W. B. (2007). Asymptotic spectral theory for nonlinear time series., The Annals of Statistics 35 1773–1801.
  • [30] Tjøstheim, D. (1994). Non-linear time series: a selective review., Scandinavian Journal of Statistics 97–130.
  • [31] Tong, H. (1990)., Non-Linear Time Series: a Dynamical System Approach. Oxford University Press.
  • [32] Volný, D. and Woodroofe, M. (2014). Quenched central limit theorems for sums of stationary processes., Statistics & Probability Letters 85 161–167.
  • [33] Whittle, P. (1954). On stationary processes in the plane., Biometrika 434–449.
  • [34] Wu, W. B. (2005). Nonlinear system theory: another look at dependence., Proceedings of the National Academy of Sciences of the United States of America 102 14150–14154.
  • [35] Wu, W. B. and Shao, X. (2004). Limit theorems for iterated random functions., Journal of Applied Probability 425–436.