Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales
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.
Hida’s white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion [1–4]. In 1988, Ito  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  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 , 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  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 ). 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 ). 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 ). The space is a nuclear space; namely, for any , there exists such that the inclusion mapping defined by is a Hilbert-Schmidt operator.
Lemma 4 (see ). 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 ). 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 ). 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 ). 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 ). 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.
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 , 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 .
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.
Conflict of Interests
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).
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