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Hitting times of decreasing sets for regenerative processes

Published online by Cambridge University Press:  14 July 2016

K. B. Athreya
K. B. Athreya*
Affiliation:
Iowa State University
*
∗ Postal address: Departments of Mathematics and Statistics, Iowa State University, Ames, IA 50011, USA.
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Abstract

Let {X(t): t ≧ 0} be a regenerative process with excursion process {Y(u): 0 ≦ u < T, T} and state space . Let An be a sequence of sets in such that pn =P(Y(u)∊An for some 0≦ u < T) → 0. Let Vn be the hitting time of An for the process X. This paper gives a variety of conditions on the excursion process Y to obtain limit theorems for Vn. Apart from obtaining an exponential limit in the positive recurrent case, i.e. ET < x, some non-exponential limits are obtained in the null recurrent case. The results are illustrated via the age process.

Keywords

EXCURSION PROCESSPOSITIVE AND NULL RECURRENCEEXPONENTIAL AND NON-EXPONENTIAL LIMITSREGULAR AND SLOW VARIATION

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 1 , March 1988 , pp. 89 - 96
Copyright
Copyright © Applied Probability Trust 1988 

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Footnotes

Research supported in part by a grant from the National Science Foundation, DMS 85 02311.

References

[1] Athreya, K. B. and Ney, P. (1978) A new approach to the limit theory of Markov chains. Trans . Amer. Math. Soc. 245, 493–501.CrossRefGoogle Scholar
[2] Cinlar, E. (1975) Introduction to Stochastic Processes. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[3] Cogburn, R. (1985) On the distribution of first passage and return times for small sets. Ann. Prob. 13, 1219–1223.Google Scholar
[4] Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. 2. Wiley, New York.Google Scholar
[5] Korolyuk, D. V. and Silvestrov, D. S. (1983) Entry times into asymptotically receding domains for ergodic Markov chains. Theory Prob. Appl. 28, 432–441.Google Scholar
[6] Korolyuk, D. V. and Silvestrov, D. S. (1984) Entry times into asymptotically receding regions for processes with semi Markov switchings. Theory Prob. Appl. 29, 558–562.Google Scholar
[7] Lindvall, T. (1976) An invariance principle for thinned random measures. Studia Sci. Math. Hungar. 11, 269–275.Google Scholar
[8] Nummelin, E. (1984) General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83, Cambridge University Press.Google Scholar
[9] Rootzen, H. (1988) Maxima and exceedances of stationary Markov chains. Adv. Appl. Prob. 20 (2).CrossRefGoogle Scholar