ASYMPTOTIC DISTRIBUTIONS FOR SUMS OF INDEPENDENT EXPONENTIALLY DISTRIBUTED RANDOM VARIABLES AND FOR THE PURE BIRTH PROCESS.
Abstract
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 < 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 x(t) is normal if alpha = or < 1/2 and non-normal if 1/2 < alpha = or < 1.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1970
- Accession Number
- AD0705637
Entities
People
- Bernard Sherman
Organizations
- Rocketdyne