Abstract
The paper considers the two-sided taboo limit process that arises when a regenerative process X is conditioned on staying out of a specified set of states (taboo set) over a long period of time. The taboo limit process after time 0 is a version of X, and the time-reversal of the taboo limit process before time 0 is regenerative with tabooed cycles having exponentially biased lengths. The cycle straddling zero has that same bias up to time 0, and is unbiased after time 0. This extends to processes regenerative in the wide sense, and to processes that only regenerate as long as they have not entered the taboo set.
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References
E. Arjas, E. Nummelin and R.L. Tweedie, Uniform limit theorems for non-singular renewal and Markov renewal processes, J. Appl. Probab. 15 (1978) 112–125.
S. Asmussen, Applied Probability and Queues (Wiley, New York, 1987).
S. Asmussen, P.W. Glynn and H. Thorisson, Stationarity detection in the initial transient problem, ACM Trans. Modelling Comput. Simulation 2 (1992) 130–157.
K.B. Athreya and P. Ney, A new approach to the limit theory of recurrent Markov chains, Trans. Amer. Math. Soc. 245 (1978) 493–501.
J.A. Bucklew, Large Deviation Techniques in Decision, Simulation, and Estimation (Wiley, New York, 1990).
P.W. Glynn and H. Thorisson, Two-sided taboo limits for Markov processes and associated perfect simulation, Stochastic Process. Appl. 91 (2001) 1–20.
P.W. Glynn and H. Thorisson, Structural characterization of taboo-stationarity for general processes in two-sided time, Stochastic Process. Appl. 102 (2002) 311–318.
J. Keilson, Markov Chain Models — Rarity and Exponentiality (Springer, New York, 1979).
P. Ney and E. Nummelin, Markov additive processes I. Eigenvalue properties and limit theorems, Ann. Probab. 15 (1987) 561–592.
E. Nummelin, A splitting technique for Harris recurrent Markov chains, Z. Wahrscheinlichkeitsth. 43 (1978) 309–318.
E. Nummelin, General Irreducible Markov Chains and Non-Negative Operators (Cambridge Univ. Press, Cambridge, 1984).
E. Nummelin and E. Arjas, A direct construction of the R-invariant measure for a Markov chain on a general state space, Ann. Probab. 4 (1976) 674–679.
E. Nummelin and R. Tweedie, Geometric ergodicity and R-positivity for general Markov chains, Ann. Probab. 6 (1978) 404–420.
E. Seneta and D. Vere-Jones, On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states, J. Appl. Probab. 3 (1966) 403–434.
H. Thorisson, Coupling, Stationarity, and Regeneration (Springer, New York, 2000).
R. Tweedie, Quasi-stationary distributions for Markov chains on a general state space, J. Appl. Probab. 11 (1974) 726–741.
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Glynn, P.W., Thorisson, H. Limit Theory for Taboo-Regenerative Processes. Queueing Systems 46, 271–294 (2004). https://doi.org/10.1023/B:QUES.0000027987.00421.80
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DOI: https://doi.org/10.1023/B:QUES.0000027987.00421.80