Journal of Statistical Physics, Vol. 65, Nos. 3/4, 1991. p. 645-672.
We use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates. Both the normal and slow approaches to infinity are studied. Fluctuations are shown to obey the central limit theorem for almost all sample-transition rates. Our results are stronger, and our proofs simpler, then those of recently published studies.