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 $M$. A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strong convergent. A type of generalized martingales associated with $M$ are introduced and their convergence theorems are established. Some applications are also shown.


Introduction
Hida's white noise analysis is essentially a theory of infinite dimensional calculus on generalized functionals of Brownian motion [9,12,14,16]. In 1988, Ito [13] 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., [1,2,4,11,15]).
Discrete-time normal martingales [18] also play an important role in many theoretical and applied fields. For example, the classical random walk (a special discretetime normal martingale) is used to establish functional central limit theorems in probability theory [5,19]. It would then be interesting to develop a theory of infinite dimensional calculus on generalized functionals of discrete-time normal martingales.
Let M = (M n ) n∈N be a discrete-time normal martingale satisfying some mild conditions. In a recent paper [20], we constructed generalized functionals of M , and introduced a transform, called the Fock transform, to characterize those functionals.
In this paper, we apply the Fock transform [20] to investigate generalized functional sequences of M . First, by using the Fock transform, we obtain a necessary and sufficient condition for a generalized functional sequence of M to be strong convergent. Then we introduce a type of generalized martingales associated with M , called M -generalized martingales, which are a special class of generalized functional sequences of M and include as a special case the classical martingales with respect to the filtration generated by M . We establish a strong-convergent criterion in terms of the Fock transform for M -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 M -generalized martingale, its strong convergence is just equivalent to its strong boundedness.
Throughout this paper, N designates the set of all nonnegative integers and Γ the finite power set of N, namely where #(σ) means the cardinality of σ as a set. In addition, we always assume that (Ω, F , P ) is a given probability space with E denoting the expectation with respect to P . We denote by L 2 (Ω, F , P ) the usual Hilbert space of square integrable complex-valued functions on (Ω, F , P ) 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.

Generalized functionals
Let M = (M n ) n∈N be a discrete-time normal martingale on (Ω, F , P ) that has the chaotic representation property and Z = (Z n ) n∈N the discrete-time normal noise associated with M (see Appendix). We define And, for brevity, we use L 2 (M ) to mean the space of square integrable functionals of M , namely which shares the same inner product and norm with L 2 (Ω, F , P ), namely ·, · and · . We note that {Z σ | σ ∈ Γ} forms a countable orthonormal basis for L 2 (M ) (see Appendix).
Lemma 2.1. [22] Let σ → λ σ be the N-valued function on Γ given by Then, for p > 1, the positive term series σ∈Γ λ −p σ converges and moreover Using the N-valued function defined by (2.3), we can construct a chain of Hilbert spaces consisting of functionals of M as follows. For p ≥ 0, we define a norm · p on L 2 (M ) through and put It is not hard to check that · p is a Hilbert norm and S p (M ) becomes a Hilbert space with · p . Moreover, the inner product corresponding to · p is given by Here Z σ , ξ means the complex conjugate of Z σ , ξ .
Lemma 2.2. [20] For each p ≥ 0, one has {Z σ | σ ∈ Γ} ⊂ S p (M ) and moreover the system {λ −p σ Z σ | σ ∈ Γ} forms an orthonormal basis for S p (M ). It is easy to see that λ σ ≥ 1 for all σ ∈ Γ. This implies that · p ≤ · q and S q (M ) ⊂ S p (M ) whenever 0 ≤ p ≤ q. Thus we actually get a chain of Hilbert spaces of functionals of M : We now put and endow it with the topology generated by the norm sequence { · p } p≥0 . Note that, for each p ≥ 0, S p (M ) is just the completion of S(M ) with respect to · p . Thus S(M ) is a countably-Hilbert space [3,8]. The next lemma, however, shows that S(M ) even has a much better property.
The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see, e.g., [3] or [8]). We mention that, by identifying L 2 (M ) with its dual, one comes to a Gel'fand triple which we refer to as the Gel'fand triple associated with M .
and moreover it serves as a basis in S(M ) in the sense that where ·, · is the inner product of L 2 (M ) and the series converges in the topology of S(M ). Denote by ·, · the canonical bilinear form on S * (M ) × S(M ), namely where Φ(ξ) means Φ acting on ξ as usual. Note that ·, · denotes the inner product of L 2 (M ), which is different from ·, · .
It is easy to to verify that, for Φ, Ψ ∈ S * (M ), Φ = Ψ if and only if Φ = Ψ. Thus a generalized functional of M is completely determined by its Fock transform. The following theorem characterizes generalized functionals of M through their Fock transforms.
and in particular Φ ∈ S * q (M ).

Convergence theorems for generalized functional sequences
Let M = (M n ) n∈N be the same discrete-time normal martingale as described in Section 2. In the present section, we apply the Fock transform (see Definition 2.2) to establish convergence theorems for generalized functionals of M .
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 [3,7,8], since S(M ) is a nuclear space (see Lemma 2.3). (iii) There exists a constant p ≥ 0 such that Φ, Φ n ∈ S * p (M ), n ≥ 1, and the sequence (Φ n ) converges to Φ in the norm of S * p (M ). Here we mention that "(Φ n ) converges strongly (resp. weakly) to Φ" means that (Φ n ) converges to Φ in the strong (resp. weak) topology of S * (M ). For details about various topologies on the dual of a countably normed space, we refer to [3,7].
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 S * (M ) is strongly convergent.
Thus (Φ n ) converges weakly to Φ in S * (M ), which together with Proposition 3.1 implies that (Φ n ) converges strongly to Φ in S * (M ).
In a similar way we can prove the following theorem, which is slightly different form Theorem 3.2, but more convenient to use. Suppose Φ n (σ) converges for all σ ∈ Γ, and moreover there are constants C ≥ 0 and p ≥ 0 such that Then there exists a generalized functional Φ ∈ S * (M ) such that (Φ n ) converges strongly to Φ.

M -generalized martingales and their convergence theorems
In this section, we first introduce a type of generalized martingales associated with M , which we call M -generalized martingales, and then we use the Fock transform to a give necessary and sufficient condition for such a generalized martingale to be strongly convergent. Some other convergence results are also obtained.
Clearly Γ n] ⊂ Γ. We use I n] to mean the indicator of Γ n] , which is a function on Γ given by where I n] mean the indicator of Γ n] as defined by (4.2).
Let F = (F n ) n≥0 be the filtration on (Ω, F , P ) generated by Z = (Z n ) n≥0 , namely The following theorem justifies Definition 4.1.
The next theorem shows that for an M -generalized martingale, its strong (weak) convergence is just equivalent to its strong (weak) boundedness.

Applications
In the last section we show some applications of our main results. Recall that the system {Z σ | σ ∈ Γ} is an orthonormal basis of L 2 (M ). Now if we write n n≥0 ⊂ L 2 (M ), and moreover Ψ (0) n n≥0 is a martingale with respect to filtration F = (F n ) n≥0 . However, Ψ Recall that [20], for two generalized functionals Φ 1 , Φ 2 ∈ S * (M ), their convolution Φ 1 * Φ 2 is defined by The next theorem provides a method to construct an M -generalized martingale through the M -generalized martingale Ψ (0) n n≥0 defined in (5.1).
Theorem 5.2. Let Φ ∈ S * (M ) be a generalized functional and define is an M -generalized martingale, and moreover it converges strongly to Φ in S * (M ).
Let F ∞ = σ(M n ; n ∈ N), the σ-field over Ω generated by M . In the literature, F ∞ -measurable functions on Ω are also known as functionals of M . Thus elements of L 2 (Ω, F ∞ , P ) can be called square integrable functionals of M . For brevity, we usually denote by L 2 (M ) the space of square integrable functionals of M , namely So, if the discrete-time normal martingale M has the chaotic representation property, then the system {Z σ | σ ∈ Γ} is actually an orthonormal basis for L 2 (M ), which is a closed subspace of L 2 (Ω, F , P ) as is known.
Remark A.1.Émery [6] called a Z-indexed process X = (X n ) n∈Z satisfying (A.2) a novation and introduced the notion of the chaotic representation property for such a process.