  In this paper, we present a fractional decomposition of the probability generating function of the innovation process of the first-order non-negative integer-valued autoregressive [INAR(1)] process to obtain the corresponding probability mass function. We also provide a comprehensive review of integer-valued time series models, based on the concept of thinning operators, with geometric-type marginals. In particular, we develop four fractional approaches to obtain the distribution of innovation processes of the INAR(1) model and show that the distribution of the innovations sequence has geometric-type distribution. These approaches are discussed in detail and illustrated through a few examples. Finally, using the methods presented here, we develop four new first-order non-negative integer-valued autoregressive process for autocorrelated counts with overdispersion with known marginals, and derive some properties of these models. 
Fractional approaches for the distribution of innovation sequence of   INAR(1) processes