The Mixing Properties of Subdiagonal Bilinear Models

  • Jeon, H. (Department of Statistics, Ewha Womans University) ;
  • Lee, O. (Department of Statistics, Ewha Womans University)
  • Received : 20100600
  • Accepted : 20100800
  • Published : 2010.09.30


We consider a subdiagonal bilinear model and give sufficient conditions for the associated Markov chain defined by Pham (1985) to be uniformly ergodic and then obtain the $\beta$-mixing property for the given process. To derive the desired properties, we employ the results of generalized random coefficient autoregressive models generated by a matrix-valued polynomial function and vector-valued polynomial function.


  1. Bhaskara Rao, M., Subba Rao, T. and Walker, A. M. (1983). On the existence of some bilinear time series models, Journal of Time Series Analysis, 4, 95–110.
  2. Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes, Annals of Probability, 20, 1714–1730.
  3. Carrasco, M. and Chen, X. (2002). Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory, 18, 17–39.
  4. Chanda (1992). Stationarity and central limit theorem associated with bilinear time series models, Journal of Time Series Analysis, 12, 301–313.
  5. Doukhan, D. (1994). Mixing: Properties and Examples, Springer-Verlag, New York.
  6. Feigin, P. D. and Tweedie, R. L. (1985). Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finitness of moments, Journal of Time Series Analysis, 6, 1–14.
  7. Giraitis, L. and Surgailis, D. (2002). ARCH-type bilinear models with double long memory, Stochastic Processes and Their Applications, 100, 275–300.
  8. Granger, C. W. J. and Andersen, A. P. (1978). An Introduction to Bilinear Time Series Models, Van-denhoeck and Ruprecht, Gottingen.
  9. Hardy, G. H., Littlewood, J. and Polya, G. (1952). Inequalities, Cambridge University Press, London.
  10. Kristensen, D. (2009). On stationarity and ergodicity of the bilinear model with applications to GARCH models, Journal of Time Series Analysis, 30, 125–144.
  11. Kristensen, D. (2010). Uniform ergodicity of a class of Markov chains with applications to time series models, Preprint.
  12. Lee, O. (2006). Stationary $\beta$-mixing for subdiagonal bilinear time series, Journal of the Korean Statistical Society, 35, 79–90.
  13. Liu, J. (1992). On Stationarity and asymptotic inference of bilinear time series models, Statistica Sinica, 2, 479–494.
  14. Liu, J. and Brockwell, P. J. (1988). On the general bilinear time series model, Journal of Applied Probability, 25, 553–564.
  15. Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability, Springer, London.
  16. Pham, D. T. (1985). Bilinear Markovian representation and bilinear models, Stochastic Processes and Their Applications, 20, 295–306.
  17. Pham, D. T. (1986). The mixing property of bilinear and generalized random coefficient autoregressive models, Stochastic Processes and their Applications, 23, 291–300.
  18. Subba Rao, T. (1981). On the theory of bilinear time series models, Journal of the Royal Statistical Society Series B, 43, 244–255.
  19. Subba Rao, T. and Gabr, M. M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models, Lecture notes in Statistics 24, Springer-Verlag, New York.
  20. Terdik, G. (1999). Bilinear stochastic models and related problems of nonlinear time series analysis; A frequency domain approach, Lecture Notes in Statistics, 142, Springer, New York.