Mixture Representations for Generalized Burr, Snedecor–Fisher and Generalized Student Distributions with Related Results

Abstract

In this paper, the representability of the generalized Student’s distribution as uniform and normal-scale mixtures is considered. It is also shown that the generalized Burr and the Snedecor–Fisher distributions can be represented as the scale mixtures of uniform, folded normal, exponential, Weibull or Fréchet distributions. New multiplication-type theorems are proven for these and related distributions. The relation between the generalized Student and generalized Burr distribution is studied. It is shown that the Snedecor–Fisher distribution is a special case of the generalized Burr distribution. Based on these mixture representations, some limit theorems are proven for random sums in which the symmetric and asymmetric generalized Student or symmetric and asymmetric two-sided generalized Burr distributions are limit laws. Also, limit theorems are proven for maximum and minimum random sums and absolute values of random sums in which the generalized Burr distributions are limit laws.