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On the fluctuation of stochastically monotone Markov chains and some applications

Published online by Cambridge University Press:  14 July 2016

Harry Cohn
Harry Cohn*
Affiliation:
University of Melbourne
*
∗ Postal address: Department of Statistics, University of Melbourne, Richard Berry Building, Parkville, VIC 3052, Australia.
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Abstract

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn+1| > a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.

Keywords

STOCHASTIC MONOTONICITYCREEPINGJUMPINGINFINITE MEANINDEPENDENCERELATIVE STABILITYVARYING ENVIRONMENTIMMIGRATION
Type
Short Communications
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 178 - 184
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research partially carried out while the author was at the University of California, San Diego.

References

[1] Barbour, A. D. and Pares, A. G. (1979) Limit theorems for the simple branching process allowing immigration II. The case of infinite offspring mean. Adv. Appl. Prob. 11, 63–72.Google Scholar
[2] Chow, Y. S. and Robbins, H. (1961) On sums of independent random variables with infinite moments and “fair” games. Proc. Nat. Acad. Sci. U.S.A. 47, 330–335.Google Scholar
[3] Cohn, H. (1981) On the convergence of stochastically monotone sequences of random variables and some applications. J. Appl. Prob. 18, 592–605.CrossRefGoogle Scholar
[4] Cohn, H. (1982) On a property related to convergence in probability and some applications to branching processes. Stoch. Proc. Appl. 12, 59–72.Google Scholar
[5] Daley, D. J. (1968) Stochastically monotone Markov chains. Z. Wahrscheinlichkeitsth. 10, 305–317.Google Scholar
[6] Fearn, D. H. (1972) Galton-Watson processes with generation dependence. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 159–172.Google Scholar
[7] Goettge, R. T. (1975) Limit theorems for the supercritical Galton-Watson processes in varying environment. Math. Biosci. 28, 171–190.CrossRefGoogle Scholar
[8] Heyde, C. C. (1970) Extension of a result by Seneta for the supercritical Galton-Watson process. Ann. Math. Statist. 41, 739–742.Google Scholar
[9] Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
[10] McPhee, I. M. (1982) On the Galton-Watson Process in Varying Environment and an Example with Infinitely Many Rates. , University of Melbourne.Google Scholar
[11] Schuh, H.-J. and Barbour, A. (1977) On the asymptotic behaviour of branching processes with infinite mean. Adv. Appl. Prob. 9, 681–723.CrossRefGoogle Scholar
[12] Seneta, E. (1975) Normed-convergence theory for supercritical branching processes. Stoch. Proc. Appl. 3, 35–43.Google Scholar
[13] Seneta, E. (1970) On the supercritical Galton-Watson process with immigration. Math. Biosci. 7, 9–14.Google Scholar