Generalized Gamma Distributions As Mixed Exponential Laws And Related Limit Theorems

A theorem due to L. J. Gleser stating that a gamma distribution with shape parameter no greater than one is a mixed exponential distribution is extended to generalized gamma distributions introduced by E. W. Stacy as a special family of lifetime distributions containing both gamma distributions, exponential power and Weibull distributions. It is shown that the mixing distribution is a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the mixed Poisson distribution with the generalized gamma mixing law as a mixed geometric distribution. Limit theorems are proved establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the mixed Poisson distribution with the generalized gamma mixing law including random sums to special normal mixtures.