Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Abstract

The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale M. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strongly convergent. A type of generalized martingales associated with M is introduced and their convergence theorems are established. Some applications are also shown.