A non-Markovian birth process with logarithmic growth
- 1 December 1975
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 12 (4) , 673-683
- https://doi.org/10.2307/3212718
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
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