#### 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 . 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 is introduced and their convergence theorems are established. Some applications are also shown.

#### 1. Introduction

Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion [1–4]. In 1988, Ito [5] introduced his analysis of generalized Poisson functionals, which can be viewed as a theory of 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. For example, the classical random walk (a special discrete-time normal martingale) is used to establish functional central limit theorems in probability theory [12, 13]. It would then be interesting to develop a theory of infinite dimensional calculus on generalized functionals of discrete-time normal martingales.

Let be a discrete-time normal martingale satisfying some mild conditions. In a recent paper [14], we constructed generalized functionals of and introduced a transform, called the Fock transform, to characterize those functionals.

In this paper, we apply the Fock transform [14] to investigate generalized functional sequences of . First, by using the Fock transform, we obtain a necessary and sufficient condition for a generalized functional sequence of to be strongly convergent. Then, we introduce a type of generalized martingales associated with , called -generalized martingales, which are a special class of generalized functional sequences of and include as a special case the classical martingales with respect to the filtration generated by . We establish a strong-convergent criterion in terms of the Fock transform for -generalized martingales. Some other convergence criteria are also obtained. Finally, we show some applications of our main results.

Our one interesting finding is that, for an -generalized martingale, its strong convergence is just equivalent to its strong boundedness.

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. In addition, we always 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.

#### 2. Generalized Functionals

Let be a discrete-time normal martingale on that has the chaotic representation property and the discrete-time normal noise associated with (see Appendix). We define the following: And, 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 . We note that forms a countable orthonormal basis for (see Appendix).

Lemma 1 (see [15]). *Let be the -valued function on given byThen, for , the positive term series converges and, moreover, *

Using the -valued function defined by (4), we can construct a chain of Hilbert spaces consisting of functionals of as follows. For , we define a norm on throughand putIt is not hard to check that is a Hilbert norm and becomes a Hilbert space with . Moreover, the inner product corresponding to is given byHere, means the complex conjugate of .

Lemma 2 (see [14]). *For each , one has and moreover the system forms an orthonormal basis for .*

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 putand 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 [16, 17]. The next lemma, however, shows that even has a much better property.

Lemma 3 (see [14]). *The space is a nuclear space; namely, for any , there exists such that the inclusion mapping defined by is a Hilbert-Schmidt operator.*

For , we denote by the dual of and by 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., [16] or [17]).

Lemma 4 (see [14]). *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 .

Lemma 5 (see [14]). *The system is contained in and moreover it serves as a basis in in the sense thatwhere is the inner product of and the series converges in the topology of .*

*Definition 6 (see [14]). *Elements of are called generalized functionals of , while elements of are called testing functionals of .

Denote by the canonical bilinear form on ; namely,where means acting on as usual. Note that denotes the inner product of , which is different from .

*Definition 7 (see [14]). *For , its Fock transform is the function on given bywhere is the canonical bilinear form.

It is easy to verify that, for , , if and only if . Thus, a generalized functional of is completely determined by its Fock transform. The following theorem characterizes generalized functionals of through their Fock transforms.

Lemma 8 (see [14]). *Let be a function on . Then, is the Fock transform of an element of if and only if it satisfies for some constants and . In that case, for , one hasand in particular .*

#### 3. Convergence Theorems for Generalized Functional Sequences

Let be the same discrete-time normal martingale as described in Section 2. In the present section, we apply the Fock transform (see Definition 7) to establish convergence theorems for generalized functionals of .

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 [16–18], since is a nuclear space (see Lemma 3).

Proposition 9. *Let , , , be generalized functionals of . Then, the following conditions are equivalent:*(i)*The sequence converges weakly to in .*(ii)*The sequence converges strongly to in .*(iii)*There exists a constant such that , , , and the sequence converges to in the norm of .*

Here, we mention that “ converges strongly (resp., weakly) to ” meaning that converges to in the strong (resp., weak) topology of . For details about various topologies on the dual of a countably normed space, we refer to [16, 18].

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 is strongly convergent.

Theorem 10. *Let , , , be generalized functionals of . Then, the sequence converges strongly to in if and only if it satisfies the following: *(1)*, for all .*(2)*There are constants and such that*

*Proof. *Regarding the “only if” part, let converge strongly to in . Then, we obviously have because and also converges weakly to . On the other hand, by Proposition 9, we know that there exists such that , , , and converges to in the norm of , which implies that . Therefore, Regarding the “if” part, let satisfy conditions (1) and (2). Then, by taking and using Lemma 8, we getin particular, , . On the other hand, is total in , which, together with (21) as well as the property implies that and Thus, converges weakly to in , which together with Proposition 9 implies that converges strongly to in .

In a similar way, we can prove the following theorem, which is slightly different from Theorem 10, but more convenient to use.

Theorem 11. *Let be a sequence of generalized functionals of . Suppose converges, for all , and moreover there are constants and such thatThen, there exists a generalized functional such that converges strongly to .*

#### 4. -Generalized Martingales and Their Convergence Theorems

In this section, we first introduce a type of generalized martingales associated with , which we call -generalized martingales, and then we use the Fock transform to give necessary and sufficient condition for such a generalized martingale to be strongly convergent. Some other convergence results are also obtained.

For a nonnegative integer , we denote by the power set of ; namely,Clearly . We use to mean the indicator of , which is a function on given by

*Definition 12. *A sequence is called an -generalized martingale if it satisfies thatwhere mean the indicator of as defined by (26).

Let be the filtration on generated by ; namely,The following theorem justifies Definition 12.

Theorem 13. *Suppose is a martingale with respect to filtration . Then, is an -generalized martingale.*

*Proof. *By the assumptions, satisfies the following conditions:where means the conditional expectation given -algebra . Note that which, together with (29) and the expansion , gives Taking Fock transforms yields where . Thus, is an -generalized martingale.

The next theorem gives a necessary and sufficient condition in terms of the Fock transform for an -generalized martingale to be strongly convergent.

Theorem 14. *Let be an -generalized martingale. Then, the following two conditions are equivalent: *(1)* is strongly convergent in .*(2)*There are constants and such that*

*Proof. *By Theorem 10, we need only to prove “”. Let be taken. Then, by the definition of -generalized martingales (see Definition 12), we have Now take such that . Then, and moreover which implies that converges. Thus, by Theorem 11, is strongly convergent in .

Theorem 15. *Let be a subset of . Then, the following two conditions are equivalent: *(1)*There is a constant such that is contained and bounded in .*(2)*There are constants and such that*

*Proof. *Obviously, condition (1) implies condition (2). We now verify the inverse implication relation. In fact, under condition (2), by using Lemma 8. we have where , which clearly implies condition (1).

The next theorem shows that, for an -generalized martingale, its strong (weak) convergence is just equivalent to its strong (weak) boundedness.

Theorem 16. *Let be an -generalized martingale. Then, the following conditions are equivalent: *(1)* is strongly convergent in .*(2)* is weakly bounded in .*(3)* is strongly bounded in .*(4)* is bounded in for some .*

*Proof. *Clearly, conditions (2), (3), and (4) are equivalent to each other because is a nuclear space (see Lemma 3). Using Theorems 14 and 15, we immediately know that conditions (1) and (4) are also equivalent.

#### 5. Applications

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

Recall that the system is an orthonormal basis of . Now, if we writethen , and moreover is a martingale with respect to filtration . However, is not convergent in since where means the cardinality of as a set and is the norm in .

Proposition 17. *The sequence defined above is an -generalized martingale, and moreover it is strongly convergent in .*

*Proof. *According to Theorem 13, is certainly an -generalized martingale. On the other hand, in viewing the relation between the canonical bilinear form on and the inner product in , we havewhich implies that with and . It then follows from Theorem 14 that is strongly convergent in .

Recall that [14], for two generalized functionals , , their convolution is defined by

The next theorem provides a method to construct an -generalized martingale through the -generalized martingale defined in (38).

Theorem 18. *Let be a generalized functional and defineThen, is an -generalized martingale, and moreover it converges strongly to in .*

*Proof. *By Lemma 8, there exist some constants and such thatOn the other hand, by using (40), we getwhich, together with the fact that , gives Thus, is an -generalized martingale. Additionally, it easily follows from (44) and (45) that , for each , and Therefore, by Theorem 11, we finally find converges strongly to .

#### Appendix

In this appendix, we provide some basic notions and facts about discrete-time normal martingales. For details, we refer to [11, 19].

Let be 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.

*Definition A.1. *A stochastic process on is called a discrete-time normal martingale if it is square integrable and satisfies the following:(i) and , for .(ii) and , for ,where , , for , and means the conditional expectation.

Let be a discrete-time normal martingale on . Then, one can construct from a process as follows:It can be verified that admits the following properties: Thus, it can be viewed as a discrete-time noise.

*Definition A.2. *Let be a discrete-time normal martingale. Then, the process defined by (A.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 A.3. *Let be a discrete-time normal martingale and 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 . For brevity, we usually denote by the space of square integrable functionals of ; namely,

*Definition A.4. *The discrete-time normal martingale is said to have the chaotic representation property if the system defined by (A.3) is total in .

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

*Remark A.5. *Émery [20] called a -indexed process satisfying (A.2) a novation and introduced the notion of the chaotic representation property for such a process.

#### Conflict of Interests

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

#### Acknowledgment

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