The purpose of this paper is to introduce and develop a family of ℤ + -valued autoregressive processes of order p ( INAR ( p ) ) by using the generalized multiplication ⊙ F of van Harn and Steutel (1982). We obtain various distributional and regression properties for these models. A number of stationary INAR ( p ) processes with specific marginals are presented and are shown to generalize several existing models.