Discrete-Continuous Dual Families, Reciprocal Laws, Random Summation, and Mixtures of Gaussian Distributions

We present a straightforward approach for constructing dual continuous-discrete families of distributions. The discrete family comprises integer-valued random variables derived through the discretization of the continuous family members. Conversely, the continuous family members are obtained as weak limits of the scaled members of the discrete family. Additionally, we introduce a novel concept of discrete reciprocal distributions and demonstrate connections to limiting distributions arising from random summation schemes and mixtures of Gaussian distributions. Several examples featuring classical continuous and discrete distributions, along with their newly derived discrete and continuous analogs, are provided to illustrate the theoretical framework. Simulation and data examples are included to further validate and demonstrate the practical relevance of the theoretical results.