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Large exceedences for uniformly recurrent Markov-additive processes and strong-mixing stationary processes

Part of: Stochastic processes Limit theorems Markov processes

Published online by Cambridge University Press:  14 July 2016

Tim Zajic
Tim Zajic*
Affiliation:
Stanford University
*
∗Present address: 422 Loma Vista, El Segundo, CA 90245, USA.
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Abstract

We extend large exceedence results for i.i.d. -valued random variables to a class of uniformly recurrent Markov-additive processes and stationary strong-mixing processes. As in the i.i.d. case, the results are proved via large deviations estimates.

Keywords

LARGE SEGMENTAL SUMSLARGE DEVIATIONSMARKOV CHAINSSTRONG MIXING

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G10: Stationary processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 679 - 691
Copyright
Copyright © Applied Probability Trust 1995 

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References

[1] Bansal, R. and Papantoni-Kazakos, P. (1986) An algorithm for detecting a change in a stochastic process. IEEE Trans. Inf. Theory 2, 227–235.CrossRefGoogle Scholar
[2] Bradley, R. C. (1983) On the ?-mixing condition for stationary random sequences. Trans. Amer. Math. Soc. 276, 55–66.Google Scholar
[3] Bradley, R. C. (1986) Basic properties of strong mixing conditions. In Dependence in Probability and Statistics, ed. Eberlein, E. and Taqqu, M., pp. 165–192. Birkhäuser, Basel.CrossRefGoogle Scholar
[4] Bryc, W. (1992) On large deviations for uniformly strong mixing sequences. Stoch. Proc. Appl. 41, 191–202.CrossRefGoogle Scholar
[5] Bryc, W. and Smolenski, W. (1993) On the convergence of averages of mixing sequences. J Theoret. Prob. 6, 473–483.CrossRefGoogle Scholar
[6] Chang, C. S. (1993) Sample path large deviations and intree networks. Queueing Systems. To appear.Google Scholar
[7] Dembo, A. and Karlin, S. (1993) Central limit theorems of partial sums for large segmental values. Stoch. Proc. Appl. 45, 259–271.CrossRefGoogle Scholar
[8] Dembo, A. and Zajic, T. (1993) Large deviations: from empirical mean and measure to partial sums process. Stoch. Proc. Appl. To appear.Google Scholar
[9] Dembo, A. and Zeitouni, O. (1993) Large Deviations Techniques and Applications. Jones and Bartlett, Boston.Google Scholar
[10] Dembo, A., Karlin, S. and Zeitouni, O. (1995) Large exceedences for multidimensional Lévy processes. Ann. Appl. Prob. To appear.CrossRefGoogle Scholar
[11] Deuschel, J. and Stroock, D. (1989) Large Deviations. Academic Press, Boston.Google Scholar
[12] Ellis, R. (1984) Large deviations for a general class of random vectors. Ann. Prob. 12, 1–12.CrossRefGoogle Scholar
[13] Hammersley, J. (1962) Generalization of the fundamental theorem on subadditive functions. Proc. Camb. Phil. Soc. 58, 1629–1639.CrossRefGoogle Scholar
[14] Iglehart, D. (1972) Extreme values in the GI/GI/1 queue. Ann. Math. Statist. 43, 627–635.CrossRefGoogle Scholar
[15] Iscoe, I., Ney, P. and Nummelin, E. (1985) Large deviations of uniformly recurrent Markov additive processes. Adv. Appl. Math. 6, 373–412.CrossRefGoogle Scholar
[16] Jensen, J. L. (1991) Saddlepoint expansions for sums of Markov dependent variables on a continuous state space. Prob. Theory Rel. Fields. 89, 181–199.CrossRefGoogle Scholar
[17] Karlin, S. and Altschul, S. F. (1990) New methods for assessing the statistical significance of molecular sequence features by using general scoring schemes. Proc. Natl. Acad. Sci. USA 87, 2264–2268.CrossRefGoogle Scholar
[18] Karlin, S. and Dembo, A. (1992) Limit distributions of maximal segmental score among Markovdependent partial sums. Adv. Appl. Prob. 24, 113–140.CrossRefGoogle Scholar
[19] Siegmund, D. (1985) Sequential Analysis: Tests and Confidence Intervals. Springer-Verlag, New York.CrossRefGoogle Scholar
[20] Sion, M. (1958) On general minimax thoerems. Pacific J. Math. 8, 171–176.CrossRefGoogle Scholar