Characterization Theorems for Generalized Functionals of Discrete-Time Normal Martingale

In this paper, we aim at characterizing generalized functionals of discrete-time normal martingales. Let $M=(M_n)_{n\in \mathbb{N}}$ be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of $M$ with an appropriate orthonormal basis for $M$'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.


Introduction
Hida's white noise analysis is essentially an infinite dimensional calculus on generalized functionals of Brownian motion [8,11,13,17]. In 1988, Y. Ito [12] 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 continuoustime processes (see, e.g., [1,2,4,10,14]).
Discrete-time normal martingales [19] also play an important role in many theoretical and applied fields [15,20]. It would then be interesting to develop an infinite dimensional calculus on generalized functionals of discrete-time normal martingales. In [22], 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 M = (M n ) n∈N be a discrete-time normal martingale that has the chaotic representation property. We first construct testing and generalized functionals of M with an appropriate orthonormal basis for M '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., [8,9,11,13,17,18]).

Discrete-time normal martingale
Throughout this paper, N designates the set of all nonnegative integers and Γ the finitie power set of N, namely where #(σ) means the cardinality of σ as a set. It is not hard to check that Γ is countable as an infinite set. Additionally, we assume that (Ω, F , P ) is a given probability space. 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.
is called a discrete-time normal martingale if it is square integrable and satisfies: Now let M = (M n ) n∈N be a discrete-time normal martingale on (Ω, F , P ). We give some necessary notions concerning M . First we construct from M a process Z = (Z n ) n∈N as It can be verified that Z admits the following properties: 3) E[Z n |F n−1 ] = 0 and E[Z 2 n |F n−1 ] = 1, n ∈ N. Thus, it can be viewed as a discrete-time noise (see [19]). The next lemma shows that, from the discrete-time normal noise Z, one can get an orthonormal system in L 2 (Ω, F , P ), which is indexed by σ ∈ Γ.
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 . So, if the discrete-time normal martingale M has the chaotic representation property, then the system {Z σ | σ ∈ Γ} defined by (2.4) is actually an orthonormal basis for L 2 (Ω, F ∞ , P ), which is a closed subspace of L 2 (Ω, F , P ) as is known.
Remark 2.1.Émery [5] called a Z-indexed process X = (X n ) n∈Z a novation, provided it satisfies (2.3), and introduced the notion of the chaotic representation property for such a process.

Generalized functionals of discrete-time nornal martingale
In the present section, we show how to construct generalized functionals of a discrete-time normal martingale.
Let M = (M n ) n∈N be a discrete-time normal martingale on (Ω, F , P ) that has the chaotic representation property. We denote by Z = (Z n ) n∈N the discrete-time normal noise associated with M (see (2.2) for its definition) and use the notation Z σ as defined in (2.4).
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 · .
Lemma 3.1. [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 (3.2), we can construct a chain of Hilbert spaces 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 .
To complete the proof, we need only to show that it is also total in S p (M ). In fact, we have So, if ξ ∈ S p (M ) satisfies that λ −p σ Z σ , ξ p = 0 for all σ ∈ Γ, then it must satisfy that Z σ , ξ = 0 for all σ ∈ Γ, which implies that ξ = 0 because the system 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 ( [3,6]). The next lemma, however, shows that S(M ) even has a much better property. Proof. Let p ≥ 0. Then there exists q > p such that 2(q − p) > 1. By Lemma 3.2, {λ −q σ Z σ | σ ∈ Γ} is an orthonormal basis for S q (M ). Thus, it follows from Lemma 3.1 that where · HS denotes the Hilbert-Schmidt norm of an operator. Therefore the inclusion mapping i pq : S q (M ) → S p (M ) is a Hilbert-Schmidt operator.
For p ≥ 0, we denote by S * p (M ) the dual of S p (M ) and · −p the norm of S * p (M ). Then S * p (M ) ⊂ S * q (M ) and · −p ≥ · −q whenever 0 ≤ p ≤ q. The lemma below is then an immediate consequence of the general theory of countably-Hilbert spaces (see, e.g., [3] or [6]).  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 .
Theorem 3.5. The system {Z σ | σ ∈ Γ} is contained in S(M ) and moreover it forms a basis for 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 ).
Proof. It follows from Lemma 3.2 and the definition of S(M ) that the system {Z σ | σ ∈ Γ} is contained in S(M ). Let ξ ∈ S(M ). Then, for each p ≥ 0, we have ξ ∈ S p (M ), which together with Lemma 3.2 gives where the series on the righthand side converges in norm · p . On the other hand, we find (3.14) λ −p σ Z σ , ξ p = λ p σ Z σ , ξ , p ≥ 0. Thus ξ = σ∈Γ Z σ , ξ Z σ and the series on the righthand side converges in · p for each p ≥ 0, namely in the topology of S(M ).

Characterization theorems
Let M = (M n ) n∈N be the same as in Section 3. In this section, we establish some characterization theorems for generalized functionals of M , 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 S * (M ) × S(M ), namely Note that ·, · denotes the inner product of L 2 (M ), which is different from ·, · . Recall that {Z σ | σ ∈ Γ} ⊂ S(M ). This allows us to introduce the following definition.
Definition 4.1. For Φ ∈ S * (Ω), its Fock transform is the function Φ on Γ given by where ·, · is the canonical bilinear form.
The theorem below shows that a generzlized functional of M is completely determined by its Fock tramsform. Proof. Clearly, we need only to prove the "if" part. To do so, we assume Φ = Ψ. Then, for each ξ ∈ S(M ), by using Theorem 3.5 and the continuity of Φ and Ψ we have  | Φ(σ)| ≤ Cλ p σ , σ ∈ Γ.
Proof. By Lemma 3.4, there exists some p ≥ 0 such that Φ ∈ S * p (M ). Now write C = Φ −p . Then, for each σ ∈ Γ, we have This completes the proof.  Proof. We first show that the series σ∈Γ Z σ , ξ F (σ) absolutely converges for each ξ ∈ S(M ), where ·, · is the inner product of L 2 (M ). In fact, by taking q > p + 1 2 , we have which, together with the following equalities and the condition described by (4.4), gives with ξ ∈ S(M ), which together with Lemma 3.1 yields that namely the series σ∈Γ Z σ , ξ F (σ) absolutely converges for each ξ ∈ S(M ). We now show that there exists a unique Φ ∈ S * (M ) such that F = Φ. In fact, we can define a functional Φ on S(M ) as Clearly, Φ is a linear functional on S(M ). Moreover, by using (4.5), we find where q > p + 1 2 . Thus Φ ∈ S * (M ). A simple calculation gives that Φ = F and, finally, Theorem 4.1 implies that such a Φ ∈ S * (M ) is unique. Theorems 4.2 and 4.3 characterize generalized functionals of M 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 Γ is the Fock transform of a generalized functional of M .  |F (σ)| ≤ Cλ p σ , σ ∈ Γ, where C ≥ 0 and p ≥ 0 are some constants independant of σ ∈ Γ.
Remark 4.1. The condition described by (4.7) is actually a type of growth condition. This corollary then shows that growth condition is enough to characterize generalized functionals of M .
Let η ∈ S(M ). Then there exists a continuous linear functional Φ η on L 2 (M ) such that η = sup |Φ η (ξ)| | ξ = 1, ξ ∈ L 2 (M ) and where ·, · and · are the inner product and norm of L 2 (M ), respectively. As a functional on S(M ), Φ η is obviously continuous with respect to the topology of S(M ), thus Φ η ∈ S * (M ). Based on these observations, we come to the next theorem, which actually offers a characterization of testing functionals of M .
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 σ∈Γ F (σ)Z σ in S(M ). Let p ≥ 0. Then we can take q > p such that 2(q − p) > 1. By the condition on F , there exists a constant C 0 ≥ 0 such that |F (σ)| ≤ C 0 λ −q σ , σ ∈ Γ, which, together with Lemma 3.1, yields On the other hand, σ∈Γ F (σ)Z σ is an orthogonal series with respect to · p . This together with (4.10) implies that it converges in · p . Thus, by the arbitrariness of the choice of p ≥ 0, it converges actually in S(M ). Now we write η = σ∈Γ F (σ)Z σ . A simple calculation gives that F (σ) = η, Z σ for σ ∈ Γ, which together with (4.8) leads to Φ η = F . Clearly, such an η ∈ S(M ) is unique.

Applications
In the last section, we show some applications of our results obtained in previous sections.
Let M = (M n ) n∈N be the same as in Section 3. We continue to use the notions and notation made in previous sections. Recall that elements of S * (M ) are called generalized functionals of M . The theorem below actually gives norm estimates to generalized functionals of M .
Theorem 5.1. Let Φ be a generalized functional of M and C ≥ 0, p ≥ 0 two constants such that Then for q > p + 1 2 one has in particular Φ ∈ S * q (M ). Proof. Let ξ ∈ S(M ). Then, by Theorem 3.5 and the continuity of Φ, we have On the other hand, by using (5.1) and (3.14), we get where σ runs over Γ. Thus which implies (5.2).
In general, the usual product of two generalized functionals of M is no longer a generalized functional of M . This means that the usual product is not a multiplication in S * (M ). The following two examples, however, show that by using our characterization theorems one can define other types of multiplication in S * (M ). We call Φ * Ψ the convolution of Φ and Ψ. It can be shown that, with * as the multiplication, S * (M ) becomes an algebra.
Remark 5.1. In [7], the authors defined the convolution for square integrable functionals of M . Here our definition of convolution actually extends that in [7].
where τ ⊂σ means that the sum is taken over all subsets of σ. We call Φ ⋄ Ψ the Wick product of Φ and Ψ.
Remark 5.2. In [22], by using the Guichardet representation, the authors defined the Wick product for generalized functionals of Bernoulli noise.