Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T12:08:57.195Z Has data issue: false hasContentIssue false
  • English
  • Français

On some distributional properties of a first-order nonnegative bilinear time series model

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Zhiqiang Zhang  and
Howell Tong
Zhiqiang Zhang*
Affiliation:
The University of Hong Kong
Howell Tong*
Affiliation:
The University of Hong Kong
*
∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong.
∗ Postal address: Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study a simple first-order nonnegative bilinear time-series model and give conditions under which the model is stationary. The probability density function of the stationary distribution (when it exists) is found. We also discuss the tail behaviour of the stationary distribution and calculate the probability density function by a numerical method. Simulation is used to check the calculation.

Keywords

Bilinear time-series modelessential upper boundessential lower boundsecond-order stationarykey renewal theoremtail behaviour

MSC classification

Primary: 60G17: Sample path properties
Secondary: 60G10: Stationary processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 3 , September 2001 , pp. 659 - 671
Copyright
Copyright © by the Applied Probability Trust 2001 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

An, H. Z. (1992). Non-negative autoregressive models. J. Time Ser. Anal 13, 283–295.Google Scholar
Andvel, J. (1989). Non-negative autoregressive processes. J. Time Ser. Anal 10, 1–11.Google Scholar
Embrechts, P., Klüppelberg, C., and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.Google Scholar
Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Prob. 1, 126–166.CrossRefGoogle Scholar
Granger, C. W., and Andersen, A. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht, Göttingen.Google Scholar
Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.CrossRefGoogle Scholar
Liu, J. (1992). On stationarity and asymptotic inference of bilinear time series models. Statist. Sinica 20, 479–494.Google Scholar
Pham, D. T. (1993). Bilinear time series models. In Dimension Estimation And Models, ed. Tong, H. World Scientific, River Edge, NJ, pp. 191–223.Google Scholar
Pourahmadi, M. (1988). Stationarity of the solution of X_t = A_tX_t-1+xe_t and analysis of non-Gaussian dependent random variables. J. Time Ser. Anal 9, 225–239.Google Scholar
Subba Rao, T. (1981). On the theory of bilinear series models. J. R. Statist. Soc 43, 244–255.Google Scholar
Tong, H. (1981). A note on a Markov bilinear stochastic process in discrete time. J. Time Ser. Anal 2, 279–284.Google Scholar
Tong, H. (1990). Nonlinear Time Series: A Dynamical System Approach. Oxford University Press.Google Scholar
Wang, S. R., An, H. Z., and Tong, H. (1983). On the distribution of a simple stationary bilinear process. J. Time Ser. Anal 4, 209–216.Google Scholar