#### Abstract

We aim at characterizing generalized functionals of discrete-time normal martingales. Let be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of with an appropriate orthonormal basis for ’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems. 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.

#### 1. Introduction

Hida’s white noise analysis is essentially an infinite dimensional calculus on generalized functionals of Brownian motion [1–4]. In 1988, Ito [5] introduced his theory of generalized Poisson functionals, which can be viewed as an infinite dimensional calculus on generalized functionals of Poisson martingale. It is known that both Brownian motion and Poisson martingale are continuous-time normal martingales. There are theories of white noise analysis for some other continuous-time processes (see, e.g., [6–10]).

Discrete-time normal martingales [11] also play an important role in many theoretical and applied fields [12, 13]. It would then be interesting to develop an infinite dimensional calculus on generalized functionals of discrete-time normal martingale. In [14], the authors defined the Wick product for generalized functionals of Bernoulli noise and analyzed its properties. In fact, generalized functionals of Bernoulli noise can be viewed as generalized functionals of a random walk.

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 be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of with an appropriate orthonormal basis for ’s square integrable functionals. Then we introduce a transform, called the Fock transform, for these functionals and characterize them via the transform. Several characterization theorems are established. Finally we give some applications of these characterization theorems.

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., [1–4, 15, 16]).

#### 2. Discrete-Time Normal Martingale

Throughout this paper, designates the set of all nonnegative integers and the finite power set of ; namely, where means the cardinality of as a set. It is not hard to check that is countable as an infinite set. Additionally, we assume that is a given probability space with denoting the expectation with respect to . We denote by the usual Hilbert space of square integrable complex-valued functions on and use and to mean its inner product and norm, respectively. By convention, is conjugate-linear in its first argument and linear in its second argument.

*Definition 1 (see [11]). *A (real-valued) stochastic process on is called a discrete-time normal martingale if it is square integrable and satisfies(i) and for ,(ii) and for ,where and for .

Now let be a discrete-time normal martingale on . We give some necessary notions concerning . First we construct from a process as It can be verified that admits the following properties: Thus, it can be viewed as a discrete-time noise (see [11]).

*Definition 2. *The process defined by (2) is called the discrete-time normal noise associated with .

The next lemma shows that, from the discrete-time normal noise , one can get an orthonormal system in , which is indexed by .

Lemma 3 (see [17, 18]). *Let be the discrete-time normal noise associated with . Define , where denotes the empty set, and **Then forms a countable orthonormal system in .*

Let , the -field over generated by . In the literature, -measurable functions on are also known as functionals of . Thus elements of can be called square integrable functionals of .

*Definition 4. *The discrete-time normal martingale is said to have the chaotic representation property if the system defined by (4) is total in .

So, if the discrete-time normal martingale has the chaotic representation property, then the system defined by (4) is actually an orthonormal basis for , which is a closed subspace of as is known.

*Remark 5. *Émery [17] called a -indexed process a novation, provided it satisfies (3), and introduced the notion of the chaotic representation property for such a process.

#### 3. Generalized Functionals of Discrete-Time Normal Martingale

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

Let be a discrete-time normal martingale on that has the chaotic representation property. We denote by the discrete-time normal noise associated with (see (2) for its definition) and use the notation as defined in (4).

For brevity, we use to mean the space of square integrable functionals of ; namely, which shares the same inner product and norm with , namely, and .

Lemma 6 (see [14]). *Let be the -valued function on given by **Then, for , the positive term series converges and moreover *

Using the -valued function defined by (6), we can construct a chain of Hilbert spaces of functionals of as follows. For , we define a norm on through and put It is not hard to check that is a Hilbert norm and becomes a Hilbert space with . Moreover, the inner product corresponding to is given by Here means the complex conjugate of .

Lemma 7. *For , one has and moreover the system forms an orthonormal basis for .*

*Proof. *For , a direct calculation gives , which means that . Clearly is an orthonormal system in . To complete the proof, we need only to show that it is also total in . In fact, we have So, if satisfies that , for all , then it must satisfy that , for all , which implies that because the system is an orthonormal basis for . Thus is total in .

It is easy to see that for all . This implies that and whenever . Thus we actually get a chain of Hilbert spaces of functionals of : We now put and endow it with the topology generated by the norm sequence . Note that, for each , is just the completion of with respect to . Thus is a countably-Hilbert space [19, 20]. The next lemma, however, shows that even has a much better property.

Lemma 8. *The space is a nuclear space; namely, for any , there exists such that the inclusion mapping defined by is a Hilbert-Schmidt operator.*

*Proof. *Let . Then there exists such that . By Lemma 7, is an orthonormal basis for . Thus, it follows from Lemma 6 that where denotes the Hilbert-Schmidt norm of an operator. Therefore the inclusion mapping is a Hilbert-Schmidt operator.

For , we denote by the dual of and the norm of . Then and whenever . The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see, e.g., [19] or [20]).

Lemma 9. *Let be the dual of and endow it with the strong topology. Then **and moreover the inductive limit topology on given by space sequence coincides with the strong topology.*

We mention that, by identifying with its dual, one comes to a Gel’fand triple which we refer to as the Gel’fand triple associated with .

Theorem 10. *The system is contained in and moreover it forms a basis for in the sense that **where is the inner product of and the series converges in the topology of .*

*Proof. *It follows from Lemma 7 and the definition of that the system is contained in . Let . Then, for each , we have , which together with Lemma 7 gives where the series on the right-hand side converges in norm . On the other hand, we find Thus and the series on the right-hand side converges in for each , namely, in the topology of .

*Definition 11. *Elements of are called generalized functionals of , while elements of are called testing functionals of .

As mentioned above, by identifying with its dual, one has the Gel’fand inclusion relation , which justifies this definition.

#### 4. Characterization Theorems

Let be the same as in Section 3. In this section, we establish some characterization theorems for testing and generalized functionals of , which are our main results.

We continue to use the notions and notation made in previous sections. Additionally, we denote by the canonical bilinear form on ; namely, Note that denotes the inner product of , which is different from .

Recall that . This allows us to introduce the following definition.

*Definition 12. *For , its Fock transform is the function on given by where is the canonical bilinear form.

The theorem below shows that a generalized functional of is completely determined by its Fock transform.

Theorem 13. *Let , . Then if and only if .*

*Proof. *Clearly, we need only to prove the “if” part. To do so, we assume . Then, for each , by using Theorem 10 and the continuity of and we have Thus .

Theorem 14. *Let . Then there exist constants and such that *

*Proof. *By Lemma 9, there exists some such that . Now write . Then, for each , we have This completes the proof.

Theorem 15. *Let be a function on satisfying **for some constants and . Then there exists a unique such that and moreover, for , one has *

*Proof. *Put where the series converges in due to the following estimate:which implies that since . Thus . Moreover we easily see that . The uniqueness of is obvious.

Theorems 14 and 15 characterize generalized functionals of through their Fock transforms. As an immediate consequence of these two theorems, we come to the next corollary, which offers a criterion for checking whether or not a function on is the Fock transform of a generalized functional of .

Corollary 16. *Let be a function on . Then is the Fock transform of an element of if and only if it satisfies **where and are some constants independent of .*

*Remark 17. *The condition described by (31) is actually a type of growth condition. This corollary then shows that growth condition is enough to characterize generalized functionals of . This 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., [1–4, 15, 16]).

Let . Then there exists a continuous linear functional on such that and where and are the inner product and norm of , respectively. As a functional on , is obviously continuous with respect to the topology of ; thus . Based on these observations, we come to the next theorem, which actually offers a characterization for testing functionals of .

Theorem 18. *Let be a function on . If satisfies that for each there exists such that **then there exists a unique such that . Conversely, if for some , then for each there exists such that (33) holds.*

*Proof. *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 in . Let . Then we can take such that . By the condition on , there exists a constant such that which, together with Lemma 6, yieldsOn the other hand, is an orthogonal series with respect to . This together with (35) implies that it converges in . Thus, by the arbitrariness of the choice of , it converges actually in .

Now we write . A simple calculation gives that , for , which together with (32) leads to . Clearly, such an is unique.

#### 5. Applications

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

Let be the same as in Section 3. We continue to use the notions and notation made in previous sections. Recall that elements of are called generalized functionals of .

*Example 19. *Consider the counting measure over . It can be shown that, as a function on , satisfies Thus, by Theorem 15, is the Fock transform of a certain generalized functional of , and moreover has a norm estimate like where .

*Example 20. *Consider the function on . Clearly, it satisfies condition (26) with and . Thus, by Theorem 15, there exists a generalized functional of , written as , such that and moreover has a norm estimate as below where .

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

Theorem 21. *Let , where . Then the norm of in satisfies *

*Proof. *By the Riesz representation theorem [21], there exists a unique such that and which, together with Lemma 7, gives This completes the proof.

In general, the usual product of two generalized functionals of is no longer a generalized functional of . This means that the usual product is not a multiplication in . The following two examples, however, show that by using our characterization theorems one can define other types of multiplication in .

*Example 22. *Let , be generalized functionals of . Then, by Theorem 14, the function satisfies condition (26). Thus there exists a unique generalized functional of , written as , such that We call the convolution of and . It can be shown that, with as the multiplication, becomes an algebra.

*Remark 23. *In [22], the authors defined the convolution for square integrable functionals of . Here our definition of convolution actually extends that in [22].

*Example 24. *Let , be generalized functionals of . Then there exists a unique generalized functional of , written as , such that where means that the sum is taken over all subsets of . We call the Wick product of and .

*Proof. *In fact, by Lemma 9, there exists such that , . Now define and take . Then, by using Theorem 21, we have which implies that , , where Thus, by Theorem 15, there exists a unique generalized functional of , written as , satisfying .

*Remark 25. *In [14], by using the Guichardet representation, the authors defined the Wick product for generalized functionals of Bernoulli noise.

#### Conflict of Interests

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

#### Acknowledgments

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).