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A new theorem on the existence of invariant distributions with applications to ARCH processes

Part of: Applications Markov processes

Published online by Cambridge University Press:  14 July 2016

Vytautas Kazakevičius  and
Remigijus Leipus
Vytautas Kazakevičius*
Affiliation:
Vilnius University
Remigijus Leipus*
Affiliation:
Vilnius University and Institute of Mathematics and Informatics
*
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania.
∗ Postal address: Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius 2600, Lithuania.
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Abstract

A new theorem on the existence of an invariant initial distribution for a Markov chain evolving on a Polish space is proved. As an application of the theorem, sufficient conditions for the existence of integrated ARCH processes are established. In the case where these conditions are violated, the top Lyapunov exponent is shown to be zero.

Keywords

Markov chaininvariant distributionintegrated ARCH processnonlinear time series

MSC classification

Primary: 60J05: Discrete-time Markov processes on general state spaces
Secondary: 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 40 , Issue 1 , March 2003 , pp. 147 - 162
Copyright
Copyright © Applied Probability Trust 2003 

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References

Baillie, R. T., Bollerslev, T., and Mikkelsen, H. O. (1996). Fractionally integrated generalized autoregressive conditional heteroskedasticity. J. Econometrics 74, 3–30.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.Google Scholar
Bougerol, P., and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52, 115–127.Google Scholar
Diaconis, P., and Freedman, D. (1999). Iterated random functions. SIAM Rev. 41, 45–76.Google Scholar
Dudley, R. M. (1989). Real Analysis and Probability. Wadsworth and Brooks/Cole, Pacific Grove, CA.Google Scholar
Edwards, R. E. (1965). Functional Analysis. Holt, Rinehart and Winston, New York.Google Scholar
Engle, R. F., and Bollerslev, T. (1986). Modelling the persistence of conditional variances. Econometric Rev. 27, 1–50.Google Scholar
Fonseca, G., and Tweedie, R. L. (2002). Stationary measures for non-irreducible non-continuous Markov chains with time series applications. Statist. Sinica 12, 651–660.Google Scholar
Giraitis, L., Kokoszka, P., and Leipus, R. (2000). Stationary ARCH models: dependence structure and central limit theorem. Econometric Theory 16, 3–22.Google Scholar
Jarner, S. F., and Tweedie, R. L. (2000). Stability properties of Markov chains defined via iterated random functions. Preprint.Google Scholar
Jarner, S. F., and Tweedie, R. L. (2001). Locally contracting iterated functions and stability of Markov chains. J. Appl. Prob. 38, 494–507.Google Scholar
Kazakevičius, V., and Leipus, R. (2002). On stationarity in the ARCH∞ model. Econometric Theory 18, 1–16.Google Scholar
Kesten, H., and Spitzer, F. (1984). Convergence in distribution of products of random matrices. Z. Wahrscheinlichkeitsth. 67, 363–386.CrossRefGoogle Scholar
Kuratowski, K., and Ryll-Nardzewski, C. (1965). A general theorem on selectors. Bull. Acad. Polon. Sci. Ser. Math. Astronom. Phys. 13, 397–403.Google Scholar
Liu, J., and Susko, E. (1992). On strict stationarity and ergodicity of a non-linear ARMA model. J. Appl. Prob. 29, 363–373.Google Scholar
Meyn, S. P., and Tweedie, R. L. (1996). Markov Chains and Stochastic Stability. Springer, London.Google Scholar
Mikosch, T. and Stărică, C. (2000). Long-range dependence effects and ARCH modelling. Preprint.Google Scholar
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1, 1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Nelson, D. B., and Cao, C. Q. (1992). Inequality constraints in the univariate GARCH model. J. Business Econom. Statist. 10, 229–235.Google Scholar
Tweedie, R. L. (1988). Invariant measures for Markov chains with no irreducibility assumptions. In A Celebration of Applied Probability (J. Appl. Prob. Spec. Vol. 25A), ed. Gani, J., Applied Probability Trust, Sheffield, pp. 275–285.Google Scholar
Tweedie, R. L. (2001). Drift conditions and invariant measures for Markov chains. Stoch. Process. Appl. 92, 345–354.Google Scholar