A new type of discrete self-decomposability and its application to continuous-time Markov processes for modeling count data time series

We propose a family of extended thinning operators, indexed by a parameter γ in [0, 1], with the boundary case of γ = 0 corresponding to the well-known binomial thinning operator. The extended thinning operators can be used to construct a class of continuous-time Markov processes for modeling count time series data. The class of stationary distributions of these processes is called generalized discrete self-decomposable, denoted by DSD (γ). We obtain characterization results for the DSD (γ) class and investigate relationships among the classes for different γ's.