We aim at characterizing generalized functionals of discrete-time normal martingales. Let

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

Discrete-time normal martingales [

In this paper, we consider a class of discrete-time normal martingales, namely, the ones that have the chaotic representation property, which include random walks, especially the classical random walk. Our main work is as follows. Let

Our results show that generalized functionals of discrete-time normal martingales can be characterized only by growth condition, which contrasts sharply with the case of some continuous-time processes (e.g., Brownian motion), where both growth condition and analyticity condition are needed to characterize generalized functionals of those continuous-time processes (see, e.g., [

Throughout this paper,

A (real-valued) stochastic process

Now let

The process

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 [

In the present section, we show how to construct generalized functionals of a discrete-time normal martingale.

Let

For brevity, we use

Let

Using the

For

For

It is easy to see that

The space

Let

For

Let

We mention that, by identifying

The system

It follows from Lemma

Elements of

As mentioned above, by identifying

Let

We continue to use the notions and notation made in previous sections. Additionally, we denote by

Recall that

For

The theorem below shows that a generalized functional of

Let

Clearly, we need only to prove the “if” part. To do so, we assume

Let

By Lemma

Let

Put

Theorems

Let

The condition described by (

Let

Let

The second part of the theorem can be proved easily. Here we only give a proof to the first part.

To do so, we consider the series

Now we write

In the last section, we show some applications of our results obtained in previous sections.

Let

Consider the counting measure

Consider the function

To show two more examples of application, we first prove a useful norm formula for elements of

Let

By the Riesz representation theorem [

In general, the usual product of two generalized functionals of

Let

In [

Let

In fact, by Lemma

In [

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

The authors are grateful to the anonymous referee for his or her valuable suggestions which improved the paper. The authors are supported by National Natural Science Foundation of China (Grant no. 11461061).