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A non-Markovian birth process with logarithmic growth

Published online by Cambridge University Press:  14 July 2016

G. R. Grimmett
G. R. Grimmett*
Affiliation:
University of Oxford
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Abstract

I show that the sum of independent random variables converges in distribution when suitably normalised, so long as the Xk satisfy the following two conditions: μ(n)= E |Xn| is comparable with E |Sn| for large n, and Xk/μ(k) converges in distribution. Also I consider the associated birth process X(t) = max{n: Sn ≦ t} when each Xk is positive, and I show that there exists a continuous increasing function v(t) such that for some variable Y with specified distribution, and for almost all u. The function v, satisfies v (t) = A (1 + o (t)) log t. The Markovian birth process with parameters λn = λn, where 0 < λ < 1, is an example of such a process.

Keywords

BIRTH PROCESSCONVERGENCE IN DISTRIBUTIONSUMS OF INDEPENDENT RANDOM VARIABLES
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 4 , December 1975 , pp. 673 - 683
Copyright
Copyright © Applied Probability Trust 1975 

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References

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