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

Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion [

Discrete-time normal martingales [

Let

In this paper, we apply the Fock transform [

Our one interesting finding is that, for an

Throughout this paper,

Let

Let

Using the

For each

It is easy to see that

The space

For

Let

We mention that, by identifying

The system

Elements of

Denote by

For

It is easy to verify that, for

Let

Let

In order to prove our main results in a convenient way, we first give a preliminary proposition, which is an immediate consequence of the general theory of countably normed spaces, especially nuclear spaces [

Let

The sequence

The sequence

There exists a constant

Here, we mention that “

The next theorem is one of our main results, which offers a criterion in terms of the Fock transform for checking whether or not a sequence in

Let

There are constants

Regarding the “only if” part, let

Regarding the “if” part, let

In a similar way, we can prove the following theorem, which is slightly different from Theorem

Let

In this section, we first introduce a type of generalized martingales associated with

For a nonnegative integer

A sequence

Let

Suppose

By the assumptions,

The next theorem gives a necessary and sufficient condition in terms of the Fock transform for an

Let

There are constants

By Theorem

Let

There is a constant

There are constants

Obviously, condition (1) implies condition (2). We now verify the inverse implication relation. In fact, under condition (2), by using Lemma

The next theorem shows that, for an

Let

Clearly, conditions (2), (3), and (4) are equivalent to each other because

In the last section, we show some applications of our main results.

Recall that the system

The sequence

According to Theorem

Recall that [

The next theorem provides a method to construct an

Let

By Lemma

In this appendix, we provide some basic notions and facts about discrete-time normal martingales. For details, we refer to [

Let

A stochastic process

Let

Let

The next lemma shows that, from the discrete-time normal noise

Let

Let

The discrete-time normal martingale

So, if the discrete-time normal martingale

Émery [

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (Grant no. 11461061).