In this talk, we define and study the notions of $k-$type proximal pairs, $k-$type asymptotic pairs and $k-$type Li Yorke sensitivity for dynamical systems given by $\mathbb{Z}^d$ actions on compact metric spaces. We prove the Auslander-Yorke dichotomy theorem for $k-$type notions. The preservation of some of these notions under conjugacy is also studied. We also study relations between these notions and their analogous notions in the usual dynamical systems.