  The objective of this paper is to introduce the notion of generalized almost
statistical (briefly, GAS) convergence of bounded real sequences, which
generalizes the notion of almost convergence as well as statistical convergence
of bounded real sequences. As a special kind of Banach limit functional, we
also introduce the concept of Banach statistical limit functional and the
notion of GAS convergence mainly depends on the existence of Banach statistical
limit functional. We prove the existence of Banach statistical limit
functional. Then we have shown the existence of a GAS convergent sequence,
which is neither statistical convergent nor almost convergent. Also, some
topological properties of the space of all GAS convergent sequences are
investigated.
