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On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations

Published online by Cambridge University Press:  01 July 2016

K. S. Chan  and
H. Tong
K. S. Chan*
Affiliation:
The Chinese University of Hong Kong
H. Tong*
Affiliation:
The Chinese University of Hong Kong
*
∗Postal address: Department of Statistics, University Science Centre, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong.
∗Postal address: Department of Statistics, University Science Centre, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong.
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Abstract

We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.

Keywords

STABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 3 , September 1985 , pp. 666 - 678
Copyright
Copyright © Applied Probability Trust 1985 

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References

Arnold, L. and Kliemann, W. (1983) Qualitative theory of stochastic systems. In Probabilistic Analysis and Related Topics 3. ed. Barucha-Reid, A. T.. Academic Press, New York.Google Scholar
Halanay, A. (1963) Quelques questions de la théorie de la stabilité pour les systèms aux differences finites. Arch. Rat. Mech. Anal. 12, 150–154.CrossRefGoogle Scholar
Kalman, R. E. and Bertram, J. E. (1960) Control system analysis and design via the “Second method” of Lyapunov II: Discrete-time systems. Trans. A.S.M.E., J. Basic Engng. D 82, 394.CrossRefGoogle Scholar
Lasalle, J. P. (1976) The Stability of Dynamical Systems. SIAM, Philadelphia, Pa.CrossRefGoogle Scholar
Nummelin, E. and Tuominen, P. (1982) Geometric ergodicity of Harris recurrent Markov chains with application to renewal theory. Stoch. Proc. Appl. 12, 187–202.CrossRefGoogle Scholar
Ozaki, T. (1980) Non-linear time series models for non-linear random vibrations. J. Appl. Prob. 17, 84–93.CrossRefGoogle Scholar
Petruccelli, J. D. and Woolford, S. W. (1984) A threshold AR (1) model. J. Appl. Prob. 21, 270–286.CrossRefGoogle Scholar
Tong, H. (1983) Threshold Models in Non-Linear Time Series Analysis. Lecture Notes in Statistics 21, Springer-Verlag, Heidelberg.CrossRefGoogle Scholar
Tweedie, R. L. (1975) Sufficient conditions for ergodicity and recurrence of Markov chains on a general state space. Stoch. Proc. Appl. 3, 385–403.CrossRefGoogle Scholar
Tweedie, R. L. (1976) Criteria for classifying general Markov chains. Adv. Appl. Prob. 8, 737–771.CrossRefGoogle Scholar
Tweedie, R. L. (1983a) Criteria for rates of convergence of Markov chains, with application to queueing theory. In Papers in Probability, Statistics and Analysis , ed, Kingman, J. F. C. and Reuter, G. E. H.. Cambridge University Press, Cambridge.Google Scholar
Tweedie, R. L. (1983b) The existence of moments for stationary Markov chains. J. Appl. Prob. 20, 191–196.CrossRefGoogle Scholar
Yoshizawa, T. (1966) Stability Theory by Liapunov’s Second Method. Publications of the Mathematical Society of Japan, No. 9.Google Scholar