ASYMPTOTIC DISTRIBUTIONS FOR SUMS OF INDEPENDENT EXPONENTIALLY DISTRIBUTED RANDOM VARIABLES AND FOR THE PURE BIRTH PROCESS.

The question of asymptotic distributions for the pure birth process is considered. For the Poisson process it is known that the state variable xt, 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 xt with parameters lambda sub k lambda k to the power alpha, k or 1, minus infinity alpha or 1, lambda a positive constant. On the basis of heuristic reasoning used in this report it is concluded that for this more general pure birth process the asymptotic distribution of the appropriately standardized state variable xt is normal if alpha or 12 and non-normal if 12 alpha or 1.