Analytic and Asymptotic Properties of the Generalized Student and Generalized Lomax Distributions

Abstract

Analytic and asymptotic properties of the generalized Student and generalized Lomax distributions are discussed, with the main focus on the representation of these distributions as scale mixtures of the laws that appear as limit distributions in classical limit theorems of probability theory, such as the normal, folded normal, exponential, Weibull, and Fréchet distributions. These representations result in the possibility of proving some limit theorems for statistics constructed from samples with random sizes in which the generalized Student and generalized Lomax distributions are limit laws. An overview of known properties of the generalized Student distribution is given, and some simple bounds for its tail probabilities are presented. An analog of the ‘multiplication theorem’ is proved, and the identifiability of scale mixtures of generalized Student distributions is considered. The normal scale mixture representation for the generalized Student distribution is discussed, and the properties of the mixing distribution in this representation are studied. Some simple general inequalities are proved that relate the tails of the scale mixture with that of the mixing distribution. It is proved that for some values of the parameters, the generalized Student distribution is infinitely divisible and admits a representation as a scale mixture of Laplace distributions. Necessary and sufficient conditions are presented that provide the convergence of the distributions of sums of a random number of independent random variables with finite variances and other statistics constructed from samples with random sizes to the generalized Student distribution. As an example, the convergence of the distributions of sample quantiles in samples with random sizes is considered. The generalized Lomax distribution is defined as the distribution of the absolute value of the random variable with the generalized Student distribution. It is shown that the generalized Lomax distribution can be represented as a scale mixture of folded normal distributions. The convergence of the distributions of maximum and minimum random sums to the generalized Lomax distribution is considered. It is demonstrated that the generalized Lomax distribution can be represented as a scale mixture of Weibull distributions or that of Fréchet distributions. As a consequence, it is demonstrated that the generalized Lomax distribution can be limiting for extreme statistics in samples with random size. The convergence of the distributions of mixed geometric random sums to the generalized Lomax distribution is considered, and the corresponding extension of the famous Rényi theorem is proved. The law of large numbers for mixed Poisson random sums is presented, in which the limit random variable has a generalized Lomax distribution.