In this paper we study with statistical convergence in the sense of the power series method which is not comparable with statistical convergence. Using this notion, we introduce the concepts of $P_{p}$--statistical exhaustiveness and weak $P_{p}$--statistical exhaustiveness. Also, we study several types of convergence of sequences of functions between two metric spaces and we obtain more general results from the concepts of exhaustiveness and the strong uniform convergence on a bornology.