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Examples for the Theory of Strong Stationary Duality with Countable State Spaces

Published online by Cambridge University Press:  27 July 2009

Persi Diaconis  and
James Allen Fill
Persi Diaconis
Affiliation:
Department of MathematicsHarvard University Cambridge, Massachusetts 02138
James Allen Fill
Affiliation:
Department of Mathematical SciencesThe Johns Hopkins University Baltimore, Maryland 21218
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Abstract

Let X1,X2,… be an ergodic Markov chain on the countable state space. We construct a strong stationary dual chain X* whose first hitting times give sharp bounds on the convergence to stationarity for X. Examples include birth and death chains, queueing models, and the excess life process of renewal theory. This paper gives the first extension of the stopping time arguments of Aldous and Diaconis [1,2] to infinite state spaces.

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Type
Articles
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Probability in the Engineering and Informational Sciences , Volume 4 , Issue 2 , April 1990 , pp. 157 - 180
Copyright
Copyright © Cambridge University Press 1990

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References

Aldous, D. & Diaconis, P. (1986). Shuffling cards and stopping times. American Mathematical Monthly 93: 333–348.CrossRefGoogle Scholar
Aldous, D. & Diaconis, P. (1987). Strong uniform times and finite random walks. Advances in Applied Mathematics 8: 69–97.CrossRefGoogle Scholar
Athreya, K.B. & Ney, P. (1978). A new approach to the limit theory of recurrent Markov chains. Transactions of the American Mathematical Society 245: 493–501.CrossRefGoogle Scholar
Beck, P. Van (1972). An application of Fourier methods to the problem of sharpening the BerryEsseen inequality. Zeitschrift für Wahrscheinlichkeitstheorie und Verwunde Gebiete 23: 187–196.CrossRefGoogle Scholar
Brown, M. (1975). The first passage time distribution for a parallel exponential system with repair. In Barlow, R.E., Fussell, J., & Singpurwalla, N. (eds)., Reliability and fault tree analysis. Conference Volume, SIAM, PA.Google Scholar
Daley, D.J. (1968). Stochastically monotone Markov chains. Zeitschrift für Wahrscheinlichkeitstheorie und Verwunde Gebiete 10: 305–317.CrossRefGoogle Scholar
Diaconis, P. (1988). Group representations in probability and statistics. Institute of Mathematical Statistics, Hayward, CA.CrossRefGoogle Scholar
Diaconis, P. & Fill, J.A. (1990). Strong stationary times via a new form of duality. Annals of Probability in press.Google Scholar
Diaconis, P. & Smith, L. (1988). Honest Bernoulli excursions. Advances in Applied Probability 25: 464–477.Google Scholar
Fill, J.A. (1990a). Strong stationary duality for continuous-time Markov chains. Unpublished manuscript.Google Scholar
Fill, J.A. (1990b). Strong stationary duality for diffusions. Unpublished manuscript.Google Scholar
Harrison, J.M. (1985). Brownian motion and stochastic flow systems. New York: Wiley.Google Scholar
Kalmykov, G. I. (1962). On the partial ordering of one-dimensional Markov processes. Theory of Probability and its Applications 7: 456–459.CrossRefGoogle Scholar
Karlin, S. & Taylor, H.M. (1981). A second course in stochastic processes. New York: Academic Press.Google Scholar
Lindvall, T. (1979). On coupling of discrete renewal processes. Zeitschrift für Wahrscheinlichkeitstheorie und Verwunde Gebiete 48: 57–70.CrossRefGoogle Scholar
Matthews, P. (1988). Some sample path properties of random walk on the cube. Journal of Theoretical Probability 2: 313–324.Google Scholar
Nagaev, S.V. (1970). On the speed of convergence of the distribution of the maximum of sums of independent random variables. Theory of Probability and its Applications 15: 309–314.CrossRefGoogle Scholar
Nevzorov, V.B. & Petrov, V.V. (1969). On the distribution of the maximum cumulative sum of independent random variables. Theory of Probability and its Applications 14: 682–687.CrossRefGoogle Scholar
Siegmund, D. (1976). The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes. Annals of Probability 4: 914–924.CrossRefGoogle Scholar
Stone, C. (1965). On characteristic functions and renewal theory. Transactions of the American Mathematical Society 120: 327–342.Google Scholar
Thorisson, H. (1988). Future independent times and Markov chains. Probability Theory and Related Fields 78: 143–148.CrossRefGoogle Scholar
Whitt, W. (1974). Heavy traffic limit theorems for queues: A survey. In Clarke, A.B. (ed), Mathematical methods in queueing theory, 307–350. Lecture notes in Economics and Mathematical Systems No. 98, Springer-Verlag, New York.CrossRefGoogle Scholar