# Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Authors: Caishi Wang; Jinshu Chen
Journal: Journal of Function Spaces
 (2015)

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


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## Body

## 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.LetM=(Mn)n∈N be a discrete-time normal martingale satisfying some mild conditions. In a recent paper [14], 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 [14] 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 strongly 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 anM-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,(1)Γ=σ∣σ⊂N,#σ<∞,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 L2(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.


## 2. Generalized Functionals

LetM=(Mn)n∈N be a discrete-time normal martingale on (Ω,F,P) that has the chaotic representation property and Z=(Zn)n∈N the discrete-time normal noise associated with M (see Appendix). We define the following: (2)Z∅=1;Zσ=∏i∈σZi,σ∈Γ,σ≠∅.And, for brevity, we use L2(M) to mean the space of square integrable functionals of M; namely,(3)L2M=L2Ω,F∞,P, which shares the same inner product and norm with L2(Ω,F,P), namely, 〈·,·〉 and ·. We note that Zσ∣σ∈Γ forms a countable orthonormal basis for L2(M) (see Appendix).Lemma 1 (see [15]).
Letσ↦λσ be the N-valued function on Γ given by(4)λσ=∏k∈σk+1,σ≠∅,σ∈Γ;1,σ=∅,σ∈Γ.Then, for p>1, the positive term series ∑σ∈Γ‍λσ-p converges and, moreover, (5)∑σ∈Γλσ-p≤exp⁡∑k=1∞k-p<∞.Using theN-valued function defined by (4), we can construct a chain of Hilbert spaces consisting of functionals of M as follows. For p≥0, we define a norm ·p on L2(M) through(6)ξp2=∑σ∈Γλσ2pZσ,ξ2,ξ∈L2Mand put(7)SpM=ξ∈L2M∣ξp<∞.It is not hard to check that ·p is a Hilbert norm and Sp(M) becomes a Hilbert space with ·p. Moreover, the inner product corresponding to ·p is given by(8)ξ,ηp=∑σ∈Γλσ2pZσ,ξ¯Zσ,η,ξ,η∈SpM.Here, 〈Zσ,ξ〉¯ means the complex conjugate of 〈Zσ,ξ〉.Lemma 2 (see [14]).
For eachp≥0, one has Zσ∣σ∈Γ⊂Sp(M) and moreover the system λσ-pZσ∣σ∈Γ forms an orthonormal basis for Sp(M).It is easy to see thatλσ≥1, for all σ∈Γ. This implies that ·p≤·q and Sq(M)⊂Sp(M) whenever 0≤p≤q. Thus, we actually get a chain of Hilbert spaces of functionals of M:(9)⋯⊂Sp+1M⊂SpM⊂⋯⊂S1M⊂S0M=L2M.We now put(10)SM=⋂p=0∞SpMand endow it with the topology generated by the norm sequence {·p}p≥0. Note that, for each p≥0, Sp(M) is just the completion of S(M) with respect to ·p. Thus, S(M) is a countably Hilbert space [16, 17]. The next lemma, however, shows that S(M) even has a much better property.Lemma 3 (see [14]).
The spaceS(M) is a nuclear space; namely, for any p≥0, there exists q>p such that the inclusion mapping ipq:Sq(M)→Sp(M) defined by ipq(ξ)=ξ is a Hilbert-Schmidt operator.Forp≥0, we denote by Sp∗(M) the dual of Sp(M) and by ·-p the norm of Sp∗(M). Then, Sp∗(M)⊂Sq∗(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., [16] or [17]).Lemma 4 (see [14]).
LetS∗(M) be the dual of S(M) and endow it with the strong topology. Then,(11)S∗M=⋃p=0∞Sp∗M and moreover the inductive limit topology on S∗(M) given by space sequence {Sp∗(M)}p≥0 coincides with the strong topology.We mention that, by identifyingL2(M) with its dual, one comes to a Gel’fand triple:(12)SM⊂L2M⊂S∗M,which we refer to as the Gel’fand triple associated with M.Lemma 5 (see [14]).
The systemZσ∣σ∈Γ is contained in S(M) and moreover it serves as a basis in S(M) in the sense that(13)ξ=∑σ∈ΓZσ,ξZσ,ξ∈SM,where 〈·,·〉 is the inner product of L2(M) and the series converges in the topology of S(M).Definition 6 (see [14]).
Elements ofS∗(M) are called generalized functionals of M, while elements of S(M) are called testing functionals of M.Denote by〈〈·,·〉〉 the canonical bilinear form on S∗(M)×S(M); namely,(14)Φ,ξ=Φξ,Φ∈S∗M,ξ∈SM,where Φ(ξ) means Φ acting on ξ as usual. Note that 〈·,·〉 denotes the inner product of L2(M), which is different from 〈〈·,·〉〉.Definition 7 (see [14]).
ForΦ∈S∗(M), its Fock transform is the function Φ^ on Γ given by(15)Φ^σ=Φ,Zσ,σ∈Γ,where 〈〈·,·〉〉 is the canonical bilinear form.It is easy to verify that, forΦ, 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.Lemma 8 (see [14]).
LetF be a function on Γ. Then, F is the Fock transform of an element Φ of S∗(M) if and only if it satisfies(16)Fσ≤Cλσp,σ∈Γ for some constants C≥0 and p≥0. In that case, for q>p+1/2, one has(17)Φ-q≤C∑σ∈Γλσ-2q-p1/2and in particular Φ∈Sq∗(M).


## 3. Convergence Theorems for Generalized Functional Sequences

LetM=(Mn)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 7) 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 [16–18], since S(M) is a nuclear space (see Lemma 3).Proposition 9.
LetΦ, Φn∈S∗(M), n≥1, be generalized functionals of M. Then, the following conditions are equivalent:(i)
The sequence(Φn) converges weakly to Φ in S∗(M).(ii)
The sequence(Φn) converges strongly to Φ in S∗(M).(iii)
There exists a constantp≥0 such that Φ, Φn∈Sp∗(M), n≥1, and the sequence (Φn) converges to Φ in the norm of Sp∗(M).Here, we mention that “(Φn) converges strongly (resp., weakly) to Φ” meaning that (Φn) 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 [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 inS∗(M) is strongly convergent.Theorem 10.
LetΦ, Φn∈S∗(M), n≥1, be generalized functionals of M. Then, the sequence (Φn) converges strongly to Φ in S∗(M) if and only if it satisfies the following: (1)
Φn^(σ)→Φ^(σ), for all σ∈Γ.(2)
There are constantsC≥0 and p≥0 such that(18)supn≥1⁡Φn^σ≤Cλσp,σ∈Γ.Proof.
Regarding the “only if” part, let(Φn) converge strongly to Φ in S∗(M). Then, we obviously have (19)Φn^σ=Φn,Zσ⟶Φ,Zσ=Φ^σ,σ∈Γ,because Zσ∣σ∈Γ⊂S(M) and (Φn) also converges weakly to Φ. On the other hand, by Proposition 9, we know that there exists p≥0 such that Φ, Φn∈Sp∗(M), n≥1, and (Φn) converges to Φ in the norm of Sp∗(M), which implies that C≡supn≥1Φn-p<∞. Therefore, (20)supn≥1⁡Φn^σ=supn≥1⁡Φn,Zσ≤supn≥1⁡Φn-pZσp=Cλσp,σ∈Γ.
Regarding the “if” part, let(Φn) satisfy conditions (1) and (2). Then, by taking q>p+1/2 and using Lemma 8, we get(21)supn≥1⁡Φn-q≤C∑σ∈Γλσ-2q-p1/2;in particular, Φn∈Sq∗(M), n≥1. On the other hand, Zσ∣σ∈Γ is total in Sq(M), which, together with (21) as well as the property (22)Φn,Zσ=Φn^σ⟶Φ^σ=Φ,Zσ,σ∈Γ,implies that Φ∈Sq∗(M) and (23)Φn,ξ⟶Φ,ξ,∀ξ∈SqM.Thus, (Φn) converges weakly to Φ in S∗(M), which together with Proposition 9 implies that (Φn) converges strongly to Φ in S∗(M).In a similar way, we can prove the following theorem, which is slightly different from Theorem10, but more convenient to use.Theorem 11.
Let(Φn)⊂S∗(M) be a sequence of generalized functionals of M. Suppose (Φn^(σ)) converges, for all σ∈Γ, and moreover there are constants C≥0 and p≥0 such that(24)supn≥1⁡Φn^σ≤Cλσp,σ∈Γ.Then, there exists a generalized functional Φ∈S∗(M) such that (Φn) converges strongly to Φ.


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

In this section, we first introduce a type of generalized martingales associated withM, which we call M-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 integern≥0, we denote by Γn the power set of {0,1,…,n}; namely,(25)Γn=σ∣σ⊂0,1,…,n.Clearly Γn⊂Γ. We use In to mean the indicator of Γn, which is a function on Γ given by(26)Inσ=1,σ∈Γn;0,σ∉Γn.Definition 12.
A sequence(Φn)n≥0⊂S∗(M) is called an M-generalized martingale if it satisfies that(27)Φn^σ=InσΦn+1^σ,σ∈Γ,n≥0,where In mean the indicator of Γn as defined by (26).LetF=(Fn)n≥0 be the filtration on (Ω,F,P) generated by Z=(Zn)n≥0; namely,(28)Fn=σZk∣0≤k≤n,n≥0.The following theorem justifies Definition 12.Theorem 13.
Suppose(ξn)n≥1⊂L2(M) is a martingale with respect to filtration F=(Fn)n≥0. Then, (ξn)n≥1 is an M-generalized martingale.Proof.
By the assumptions,(ξn)n≥1 satisfies the following conditions:(29)ξn=Eξn+1∣Fn,n≥0,where E·∣Fn means the conditional expectation given σ-algebra Fn. Note that (30)EZτ∣Fn=InτZτ,τ∈Γ,which, together with (29) and the expansion ξn+1=∑τ∈Γ‍〈Zτ,ξn+1〉Zτ, gives (31)ξn=Eξn+1∣Fn=∑τ∈ΓZτ,ξn+1EZτ∣Fn=∑τ∈ΓZτ,ξn+1InτZτ.Taking Fock transforms yields (32)ξn^σ=∑τ∈Γξn+1,ZτInτZτ^σ=ξn+1,ZσInσ=Inσξn+1^σ,where σ∈Γ. Thus, (ξn)n≥1 is an M-generalized martingale.The next theorem gives a necessary and sufficient condition in terms of the Fock transform for anM-generalized martingale to be strongly convergent.Theorem 14.
Let(Φn)n≥1⊂S∗(M) be an M-generalized martingale. Then, the following two conditions are equivalent: (1)
(Φn)n≥1 is strongly convergent in S∗(M).(2)
There are constantsC≥0 and p≥0 such that(33)supn≥1⁡Φn^σ≤Cλσp,σ∈Γ.Proof.
By Theorem10, we need only to prove “(2)⇒(1)”. Let σ∈Γ be taken. Then, by the definition of M-generalized martingales (see Definition 12), we have (34)Φm^σ=ImσΦm+k^σ,m,k≥0.Now take n0≥0 such that σ∈Γn0. Then, In0(σ)=1 and moreover (35)Φn0^σ=In0σΦn^σ=Φn^σ,n>n0,which implies that (Φn^(σ)) converges. Thus, by Theorem 11, (Φn)n≥1 is strongly convergent in S∗(M).Theorem 15.
LetD be a subset of S∗(M). Then, the following two conditions are equivalent: (1)
There is a constantp≥0 such that D is contained and bounded in Sp∗(M).(2)
There are constantsC≥0 and p≥0 such that(36)supΦ∈D⁡Φ^σ≤Cλσp,σ∈Γ.Proof.
Obviously, condition (1) implies condition (2). We now verify the inverse implication relation. In fact, under condition (2), by using Lemma8. we have (37)supΦ∈D⁡Φ-q≤C∑σ∈Γλσ-2q-p1/2,where q>p+1/2, which clearly implies condition (1).The next theorem shows that, for anM-generalized martingale, its strong (weak) convergence is just equivalent to its strong (weak) boundedness.Theorem 16.
Let(Φn)n≥1⊂S∗(M) be an M-generalized martingale. Then, the following conditions are equivalent: (1)
(Φn)n≥1 is strongly convergent in S∗(M).(2)
(Φn)n≥1 is weakly bounded in S∗(M).(3)
(Φn)n≥1 is strongly bounded in S∗(M).(4)
(Φn)n≥1 is bounded in Sp∗(M) for some p≥0.Proof.
Clearly, conditions (2), (3), and (4) are equivalent to each other becauseS(M) 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 systemZσ∣σ∈Γ is an orthonormal basis of L2(M). Now, if we write(38)Ψn0=∑τ∈ΓnZτ,n≥0,then (Ψn(0))n≥0⊂L2(M), and moreover (Ψn(0))n≥0 is a martingale with respect to filtration F=(Fn)n≥0. However, (Ψn(0))n≥0 is not convergent in L2(M) since(39)Ψn0=#Γn=2n+1/2⟶∞asn⟶∞, where #(Γn) means the cardinality of Γn as a set and · is the norm in L2(M).Proposition 17.
The sequence(Ψn(0))n≥0 defined above is an M-generalized martingale, and moreover it is strongly convergent in S∗(M).Proof.
According to Theorem13, (Ψn(0))n≥0 is certainly an M-generalized martingale. On the other hand, in viewing the relation between the canonical bilinear form on S∗(M)×S(M) and the inner product in L2(M), we have(40)Ψn0^σ=Ψn0,Zσ=Ψn0,Zσ=Inσ,σ∈Γ,n≥0,which implies that (41)supn≥0⁡Ψn0^σ≤Cλσp,σ∈Γwith C=1 and p=0. It then follows from Theorem 14 that (Ψn(0))n≥0 is strongly convergent in S∗(M).Recall that [14], for two generalized functionals Φ1, Φ2∈S∗(M), their convolution Φ1∗Φ2 is defined by(42)Φ1∗Φ2^σ=Φ1^σΦ2^σ,σ∈Γ.The next theorem provides a method to construct anM-generalized martingale through the M-generalized martingale (Ψn(0))n≥0 defined in (38).Theorem 18.
LetΦ∈S∗(M) be a generalized functional and define(43)Φn=Ψn0∗Φ,n≥0.Then, (Φn)n≥0 is an M-generalized martingale, and moreover it converges strongly to Φ in S∗(M).Proof.
By Lemma8, there exist some constants C≥0 and p≥0 such that(44)Φ^σ≤Cλσp,σ∈Γ.On the other hand, by using (40), we get(45)Φn^σ=Ψn0^σΦ^σ=InσΦ^σ,σ∈Γ,n≥0,which, together with the fact that In(σ)In+1(σ)=In(σ), gives (46)Φn^σ=InσΦn+1^σ,σ∈Γ,n≥0.Thus, (Φn)n≥0 is an M-generalized martingale. Additionally, it easily follows from (44) and (45) that Φn^(σ)→Φ^(σ), for each σ∈Γ, and (47)supn≥0⁡Φn^σ=supn≥0⁡InσΦ^σ≤Cλσp,σ∈Γ.Therefore, by Theorem 11, we finally find (Φn)n≥0 converges strongly to Φ.



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