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Tail behaviour of the stationary density of general non-linear autoregressive processes of order 1

Part of: Markov processes Inference from stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jean Diebolt  and
Dominique Guégan
Jean Diebolt*
Affiliation:
Université Paris VI
Dominique Guégan*
Affiliation:
Université Paris XIII
*
∗∗ Postal address: L.S.T.A. University Paris VI, 4 place Jussieu, Tour 45–55, 75252 Paris CEDEX 05, France.
∗∗∗ Postal address: Institut Galilée, Dept of Mathematics, University Paris XIII, Av. J.B. Clément, 93430 Villetaneuse, France.
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Abstract

We examine the main properties of the Markov chain Xt = T(Xt– 1) + σ(Xt– 1)ε t. Under general and tractable assumptions, we derive bounds for the tails of the stationary density of the process {Xt} in terms of the common density of the ε t's.

Keywords

MARKOV CHAINTAIL OF THE STATIONARY DENSITYNON-LINEAR TIME SERIES

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993 , pp. 315 - 329
Copyright
Copyright © Applied Probability Trust 1993 

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References

Broniatowski, M. (1990) On the estimation of the Weibull tail coefficient. Technical report No 118, L.S.T.A., Paris VI.Google Scholar
Diebolt, J. (1991) Estimates of the tail of the stationary density function for certain nonlinear autoregressive processes. J. Theoret. Prob. 4, 655–667.CrossRefGoogle Scholar
Diebolt, J., and Guegan, D. (1990) Probabilistic properties of the general nonlinear autoregressive process of order one. Applications to time series modelling. Technical report No 125, L.S.T.A., Paris VI.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Economica 50, 987–1007.Google Scholar
Högnas, G. (1986) Comparison of nonlinear autoregressive processes. J. Time Series Analysis 7, 205–211.Google Scholar
Granger, C.W.J. and Andersen, A.P. (1978) An Introduction to Bilinear Time Series Analysis. Vandenhoeck and Ruprecht, Göttingen.Google Scholar
Leadbetter, M.R., Lindgreen, G., and Rootzen, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag, New York.Google Scholar
Mokkadem, A. (1987) Sufficient conditions of geometric mixing for polynomial autoregressive processes – application to ARMA processes and to bilinear processes. C. R. Acad. Sci. Paris Ser. 1305, 477–480 (in French).Google Scholar
Nelson, D. (1988) Conditional heteroscedacity in asset return: a new approach. Technical report, University of Chicago.Google Scholar
Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with applications to renewal theory. Stoch. Proc. Appl. 12, 187–202.CrossRefGoogle Scholar
Ozaki, T. (1979) Nonlinear time series models for nonlinear random vibrations. Technical report, University of Manchester.Google Scholar
Tj⊘stheim, D. (1990) Non-linear time series and Markov chains. Adv. Appl. Prob. 22, 587–611.Google Scholar
Tong, H. (1990) Nonlinear Time Series. A Dynamical System Approach. Oxford University Press.Google Scholar
Tong, H. and Lim, K.S. (1980) Threshold autoregression, limit cycles and cyclical data. J.R. Statist. Soc. B 42, 245–292.Google Scholar
Tweedie, R.L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.Google Scholar
Tweedie, R.L. (1983) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar