Analogs of Gleser’s theorem for negative binomial and generalized gamma distributions and some of their applications

It is proved that the negative binomial distributions with the shape parameter less than one are mixed geometric distributions. The mixing distribution is written out explicitly. Thus, the similar result of L. Gleser, stating that the gamma distributions with the shape parameter less than one are mixed exponential distributions, is transferred to the discrete case. An analog of Gleser's theorem is also proved for generalized gamma distributions. For mixed binomial distributions related to the negative binomial laws with the shape parameter less than one, the case of a small probability of success is considered and an analog of the Poisson theorem is proved. The representation of the negative binomial distributions as mixed geometric laws is used to prove limit theorems for negative binomial random sums of independent identically distributed random variables, in particular, analogs of the law of large numbers and the central limit theorem. Both cases of light and heavy tails are considered. The expressions for the moments of limit distributions are obtained. The obtained alternative equivalent mixture representations of the limit laws provide better understanding of how mixed probability (Bayesian) models are formed.