Asymptotic Distributions for Sums of Independent Exponentially Distributed Random Variables and for the Pure Birth Process

摘要

The question of asymptotic distributions for the pure birth process is considered. For the Poisson process it is known that the state variable x(t), appropriately standardized, converges in distribution to the normal distribution. For the Yule-Furry process the asymptotic distribution is exponential. The Poisson and Yule-Furry processes are the special cases, corresponding to alpha = 0 and alpha = 1, of the pure birth process x(t) with parameters lambda sub k = lambda (k to the power alpha), k = or > 1, minus infinity