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

TL;DR

This paper uses the Fock transform to establish necessary and sufficient conditions for the strong convergence of generalized functional sequences of discrete-time normal martingales, introducing new types of generalized martingales and their convergence theorems.

Contribution

It provides a novel application of the Fock transform to characterize convergence of generalized functional sequences of discrete-time normal martingales and introduces related generalized martingales.

Findings

Established necessary and sufficient conditions for strong convergence.

Introduced new types of generalized martingales associated with the martingale.

Proved convergence theorems and demonstrated applications.

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 strong convergent. A type of generalized martingales associated with $M$ are introduced and their convergence theorems are established. Some applications are also shown.