Discrete-continuous dual families and limiting distributions of random sums: the Pareto-Normal case

We explore the framework of location-scale mixtures of Gaussian distributions (SMGD) and consider a special case where the conditional mean and variance follow the Pareto Type II (Lomax distribution). We investigate the fundamental properties of this model and its potential applications, particularly in handling heavy-tailed data. The proposed conditionally Gaussian hierarchical stochastic model provides a generalization of the Laplace probability distribution, which has already demonstrated its utility in various scientific disciplines. We present the model's basic properties and delve into related computational challenges, particularly those involving the inferential aspects of the model. The emergence of these distributions is explained as the limiting laws of suitably normalized random sums of independent and identically distributed (IID) random variables, where the number of terms follows a discrete Pareto distribution. We generalize this model by introducing a family of integer-valued random variables indexed by a parameter, which converge to infinity as the parameter becomes small. We further elaborate on constructing this family for a wide range of random variables with continuous distributions on the positive half-line. In the process, we introduce the concept of a "dual family", which encompasses both continuous and discrete distributions. The discrete members of the family are obtained by discretizing the continuous ones, and the continuous members are weak limits of the scaled discrete members. We also propose a new notion of discrete reciprocal distribution and discuss its connection to limiting distributions in the random summation scheme and mixtures of Gaussian distributions. Throughout the study, we provide several examples involving classical continuous and discrete distributions, as well as their new counterparts, to illustrate the theoretical constructions discussed. In particular, we introduce continuous analogues of well-known classical distributions like Poisson, binomial, and negative binomial, presenting basic facts related to these new models. Additionally, new results and generalizations concerning these distributions are presented, highlighting potential applications. This work also delves into computational issues that may arise when implementing these models practically, given the non-explicit nature of some of their basic characteristics. We shed light on the challenges and suggest potential solutions.In summary, this research presents a novel and comprehensive investigation into the location-scale mixtures of Gaussian distributions with Pareto Type II conditional mean and variance. The proposed model offers a powerful tool for handling heavy-tailed data and has potential applications in various scientific fields. The theoretical constructions and findings in the thesis contribute valuable insights to the understanding and application of these distributions