GG-mixed Poisson distributions as mixed geometric laws and related limit theorems
Victor Yu. Korolev, Alexander I. Zeifman · arXiv (Cornell University) · 2017
A new class of discrete GG-mixed Poisson distributions is considered as the family of mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma (GG) distributions. The latter was introduced by E. W. Stacy as a special family of lifetime distributions containing gamma, exponential power and Weibull distributions. It is proved that a generalized gamma distribution with shape and exponent power parameters no greater than one is a mixed exponential distribution. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GG-mixed Poisson distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved for random sums of independent random variables in which the number of summands has the GG-mixed Poisson distribution and the summands have both light- and heavy-tailed distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Limit theorems are proved establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GG-mixed Poisson distribution to special normal mixtures.