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Limit theorems for processes such as the Markov branching process

Published online by Cambridge University Press:  14 July 2016

Andrew D. Barbour
Andrew D. Barbour*
Affiliation:
University of Cambridge
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Abstract

Let X(t) be a continuous time Markov process on the integers such that, if σ is a time at which X makes a jump, X(σ)– X(σ–) is distributed independently of X(σ–), and has finite mean μ and variance. Let q(j) denote the residence time parameter for the state j. If tn denotes the time of the nth jump and Xn ≡ X(tb), it is easy to deduce limit theorems for from those for sums of independent identically distributed random variables. In this paper, it is shown how, for μ > 0 and for suitable q(·), these theorems can be translated into limit theorems for X(t), by using the continuous mapping theorem.

Keywords

LIMIT THEOREMSMARKOV BRANCHING PROCESSCONTINUOUS MAPPING
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 2 , June 1975 , pp. 289 - 297
Copyright
Copyright © Applied Probability Trust 1975 

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References

[1] Billingsley, P. (1968) Convergence of Probability Measures. John Wiley, New York.Google Scholar
[2] Leslie, J. R. (1973) Some limit theorems for Markov branching processes. J. Appl. Prob. 10, 299–306.CrossRefGoogle Scholar
[3] Kendall, D. G. (1966) Branching processes since 1873. J. London Math. Soc. 41, 385–406.CrossRefGoogle Scholar
[4] Stone, C. (1963) Limit theorems for random walks, birth and death processes, and diffusion processes. Illinois J. Math. 7, 638–660.CrossRefGoogle Scholar
[5] Strassen, V. (1964) An invariance principle for the law of the iterated logarithm. Z. Wahrscheinlichkeitsth. 3, 211–226.CrossRefGoogle Scholar
[6] Whitt, W. (1972) Stochastic Abelian and Tauberian theorems Z. Wahrscheinlichkeitsth. 22, 251–267.CrossRefGoogle Scholar