# Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials

Authors: Yoon Tae Kim; Hyun Suk Park
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov tripleE,μ,Γ, where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.


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## Body

## 1. Introduction

The aim of this paper is to find the fourth moment bound in the normal approximation of a random variable related to a general Markov diffusion generator. A central limit theorem, known as the fourth moment theorem, was first discovered in [1] by Nualart and Peccati, where the authors found a necessary and sufficient condition such that a sequence of random variables, belonging to a fixed Winer chaos, converges in distribution to a Gaussian random variable.Throughout this paper, we use the mathematical expectation for the integral on a probability spaceΩ,F,ℙ, so that, for example, the integral ∫ΩFdℙ is denoted by EF, and a real-valued measurable function defined on a probability space will be called a random variable. Also, we define the variance of a random variable F in L2Ω,ℙ as
(1)VarF=∫ΩF2dℙ−∫ΩFdℙ2.Theorem 1 (fourth moment theorem).
Fix an integerq≥2, and consider a sequence of random variables Fn,n≥1 belonging to the qth a Wiener chaos with EFn2=1 for all n≥1. Then, Fn⟶LZ if and only if EFn4⟶3, where Z is a standard Gaussian random variable and the notation ⟶L denotes the convergence in distribution.
Such a result constitutes a dramatic simplification of the method of moments and cumulants. In the paper [2], the fourth moment theorem is expressed in terms of Malliavin derivative. However, the results given in [1, 2] do not provide any estimates, whereas the authors in [3] prove that Theorem 1 can be recovered from the estimate of the Kolmogorov (or total variation, Wasserstein) distance obtained by using the techniques based on the combination between Malliavin calculus (see, e.g., [4–6]) and Stein’s method for the normal approximation (see, e.g., [7–9]). Also, we refer to the papers [3, 4, 10–13] for an explanation of these techniques.
For estimates for a normal approximation, we consider the Kolmogorov distances of the type(2)dKolX,Z=supz∈ℝℙX≤z−ℙZ≤z,where Z is a standard Gaussian random variable. The following statement is the remarkable achievement of Nourdin-Peccati [3] approach (see Theorem 3.1 in [3]). Let F∈D1,2 be such that EF=0 and EF2=1. Then, the following bound holds:
(3)dKolF,Z≤E1−Γ1F2,where Γ1F=DF,−DL−1FH. The notations D1,2, D, and L−1, related to Malliavin calculus, are explained in [5] or [6]. In the particular case where F is an element in the qth Wiener chaos of X with EF2=1, the upper bound in (3) is given by
(4)dKolF,Z≤q−13qEF4−3.
Here,1/6EF4−3 is just the fourth cumulant κ4F of F.Recently, the author in [14] proves, from a purely spectral point of view, that the fourth moment theorem also holds in the general framework of Markov diffusion generators. Precisely, under a certain spectral condition on a Markov diffusion generator, a sequence of eigenfunctions of such a generator satisfies the bound given in (4) with a different constant. In particular, this new technique avoids the use of complicated product formula. The authors in [15] introduce a Markov chaos of eigenfunctions being less restrictive than the notion of Markov chaos defined in [14] and also obtain the quantitative four moment theorem for convergence of the eigenfunctions towards Gaussian, gamma, and beta distributions. Furthermore, Bourguin et al. in [16] prove that convergence of the elements of a Markov chaos to a Pearson distribution can be still bounded with just the first four moments of the form
(5)dG,Z≤cξ1∫EPGdμ+ξ2∫EQGdμ,where d is a suitable distance, Z is a random variable with the law belonging to the Pearson family, and G is a chaotic random variable defined on E,μ. Here, P and Q in (5) are polynomials of degree four, and the constants ξ1 and ξ2 are determined in terms of a chaos grade defined in Definition 3.5 of [16].In this paper, we find a bound of the form(6)dKolF,Z≤ξEF4−3EF22for the Kolmogorov distance between a stand Gaussian random variable Z and a random variable F defined on E,μ,Γ related to a Markov diffusion generator L with an invariant measure μ. Since usually the central limit theorem (normal approximation) is a main topic of convergence in distribution, we confine our interest in a normal approximation. In the line of this research, the motivations and contributions of our work in comparison with other works will be summarized below:
(i)
Compared to previous works [14–16], our studies are not limited to an element belonging to a fixed eigenspace of Markov chaos. Our result is a remarkable extension in comparison with other works dealing with only random variables in a fixed eigenspace. To achieve our goal, the starting point is the following bound(7)dKolF,Z≤VarΓF,−L−1F,where L−1 is the pseudo-inverse of the underlying Markov generatorL and Γ is the carré du champ operator, which is the result of the study in [16]. However, in order to find the fourth moment bound (6), we introduce a new technique relying on two types of the operators given in [17, 18]. First, we prove that the right-hand side of (7) can be represented as the sum of two integrals with the operators mentioned above, and use this representation to prove that the fourth moment bound (6) holds. This is the innovation point of this work
(ii)
If theupper chaos grade of F is strictly greater than two, then, the constant ξ2 and QG in the bound (5) are given as follows: ξ2>0 and QG=VarG2. This fact means that the fourth moment theorem in Theorem 1 is not working. However, applying the technique developed in this paper can eliminate the second term in (5), which means that, in such a random variable G, the fourth moment theorem holds unlike the previous result in [16], where the upper bound (5) for a sequence Fn of chaotic random variables is given in (111) of Remark 12(iii)
In this paper, another notion of chaos grade, called alower chaos grade, is introduced and used to provide a better estimate than the previous one obtained from (5) in [16]. Throughout this paper, the existing chaos grade in Definition 3.5 of [16] will be called an upper chaos gradeThe rest of the paper is organized as follows: Section 2 introduces some basic notations and reviews the results of Markov diffusion generator. In Section3, a new notion of chaos grade in a finite sum of Markov chaos is defined, and the orthogonal polynomials will be considered in order to illustrate the concept on chaos grades. In Section 4, our main result is covered in Theorem 8. Finally, as an application of our main result, in Section 5, we derive upper bounds in the Kolmogorov distance for an eigenfunction belonging to a fixed Markov chaos.


## 2. Preliminaries

In this section, we recall some basic facts about Markov diffusion generator. The reader is referred to [19] for a more detailed explanation. We begin by the definition of Markov triple E,μ,Γ in the sense of [19]. For the infinitesimal generator L of a Markov semigroup P=Ptt≥0 with L2μ-domain DL, we associate a bilinear form Γ. Assume that we are given a vector space A0 of DL such that for every F,G of random variables defined on a probability space E,F,μ, the product FG is in DL (A0 is an algebra). On this algebra A0, the bilinear map (carré du champ operator) Γ is defined by
(8)ΓF,G=12LFG−FLG−GLF,for every F,G∈A0×A0. As the carré du champ operator Γ and the measure μ completely determine the symmetric Markov generator L, we will work throughout this paper with Markov triple E,μ,Γ equipped with a probability measure μ on a state space E,F and a symmetric bilinear map Γ:A0×A0 such that ΓF,F≥0.Next, we construct the domainDE of the Dirichlet form E by completion of A0 and then obtain, from this Dirchlet domain, the domain DL of L. Recall the Dirchlet form E as
(9)EF,G=EΓF,GforF,G∈A0×A0.IfA0 is endowed with the norm
(10)FE=EF2+EF,F1/2,the completion of A0 with respect to this norm turns it into a Hilbert space embedded in L2μ. Once the Dirchlet domian DE is constructed, the domaion DL⊆DE is defined as all elements F∈DE such that, for all G∈DE,
(11)EF,G≤cFEG2,where cF is a finite constant only depending on F. On these domains, a relation of L and Γ holds, namely, the integration by parts formula
(12)EΓF,G=−EFLG=−EGLF.By the integration by parts formula (12) and ΓF,F≥0, the operator −L is nonnegative and symmetric, and therefore, the spectrum of −L is contained S⊆0,∞.A full Markov triple is a standard Markov triple for which there is an extended algebraA0⊂A, with no requirements of integrability for elements of A, satisfying the requirements given in Section 3.4.3 of [19]. In particular, the diffusion property holds: for any C∞ function Ψ;ℝk⟶ℝ and F1,⋯,Fk,G∈A,
(13)ΓΨF1,⋯Fk,G=∑i=1k∂iΨF1,⋯FkΓFi,G,LΨF1,⋯Fk=∑i=1k∂iΨF1,⋯FkLFi+∑i,j=1k∂ijΨF1,⋯FkΓFi,Fj.We also define the operatorL−1, called the pseudoinverse of L, satisfying for any F∈DL,
(14)LL−1F=L−1LF=F−EF.


## 3. Chaos Grade and Orthogonal Polynomials

### 3.1. Chaos Grade

Fix a probability spaceE,F,μ. We assume that −L has a discrete spectrum Λ=0=λ0<λ1<⋯<λk<⋯. Obviously, the zero is always an eigenfunction such that −L1=0. By the assumption on the spectrum of L, one has that
(15)L2E,F,μ=⊕ℓ=0∞KerL+λℓId.Now, we define chaotic random variables as follows.Definition 2.
Suppose thatF∈⊕i=1MKerL+λℓiId⊕KerL, where 0<λℓ1<λℓ2<⋯<λℓM and Fℓi∈KerL+λℓiId for i=1,⋯,M. The random variable F is called chaotic if there exist u>1 and g<1 such that uλℓM∈Λ and gλℓM∈Λ satisfy
(16)F2∈KerL+gλℓMId⊕⋯⊕KerL+uλℓMId⊕KerL.In this case, the largest numberg satisfying (16) is called the lower chaos grade of F. On the other hand, the smallest number u satisfying (16) is called the upper chaos grade of F.Remark 3.
(1)
The authors in [16] define the chaos grade of F, corresponding to the upper chaos grade in Definition 2, in the case where F is an eigenfunction with respect to an eigenvalue of the generator L. In this paper, we introduce the lower chaos grade of F, which will be used to obtain a better estimate for the four moments theorem than the estimate given in Theorem 3.9 of [16] in the particular case where the target distribution is a standard Gaussian distribution(2)
LetFℓk∈KerL+λℓkId for k∈1,⋯,M. If F is a chaotic random variable, then FℓiFℓj, i,j∈1,⋯,M, can be expanded as a sum of eigenfunctions with the eigenvalue of the largest magnitude λmaxℓi,ℓj and the eigenvalue of the smallest magnitude λminℓi,ℓj>0, i.e.,(17)FℓiFℓj∈KerL+λminℓi,ℓjId⊕⋯⊕KerL+λmaxℓi,ℓjId⊕KerL
From (17), it follows that
(18)F2∈KerL+λminId⊕⋯⊕KerL+λmaxId⊕KerL,where λmin=min1≤i,j≤Mλminℓi,ℓj and λmax=max1≤i,j≤Mλmaxℓi,ℓj.
(3)
In the paper [16], the authors describe how the a chaos grade, corresponding to the upper chaos grade in our works, behaves under tensorization. Let Li be a generator with invariant measure μi. Define a generator L=⊗i=1MLℓi by(19)L⊗i=1MFℓi=∑i=1MFℓ1×⋯×Fℓi−1×LℓiFℓi×Fℓi+1×⋯×FℓM
IfLiFi=λiFi, then F=⊗i=1MFℓi is an eigenfunction of L with eigenvalue λ=∑i=1Mλℓi. Suppose that Fℓi, 1≤i≤M, has the lower chaos grades gℓi. Then, the lower chaos grade g of F=⊗i=1MFℓi is given by
(20)g=∑i=1Mgℓiλℓi∑i=1Mλℓi.See Corollary 4.1 in [16] for the upper chaos grade of F.Next, we consider a finite dimensional eigenfunction and a finite sum of eigenfunctions to illustrate the concept on chaos grades.


### 3.2. Ornstein-Uhlenbeck Operator

We consider thed-dimensional Ornstein-Uhlenbeck generator L, defined for any test function f by
(21)Lfx=ΔfX−∑i=1dxi∂ifxforx∈ℝd,action on L2ℝd,⊗i=1dμidxi, where
(22)μidxi=12πe−xi2/2dxi.For a multi-indexm=m1,⋯,md, we define a d-dimensional Hermite polynomial by
(23)Hmx=∏i=1dHmixi.We write∣m∣=∑i=1dmi, m!=∏i=1dmi!, and m≤n if mi≤ni for all i=1,⋯,d. Let us set
(24)F=∑m≥0∣m∣=qamHmx,q≥2,where Hmi, mi=0,1,⋯, denotes the Hermite polynomial of order mi. Then, we have that F∈KerL+qId. By the well-known product formula of Hermite polynomials and a change of variables, we have that
(25)F2=∑m≥0m=q∑n≥0n=qaman∑r≤m∧nr!mrnrHm+n−2rx=∑m≥0m=q∑n≥0n=qCm,n,m∧nHm+n−2m∧nx+⋯⋯+Cm,n,0Hm+nx,where
(26)Cm,n,r=amanr!mrnr.Sincem+n−2m∧n=m1−n1,⋯,md−nd, F2 can be expanded as a sum of eigenfunctions with the smallest eigenvalue, among positive eigenvalues, being given by
(27)λmin=minm≥0,n≥0∑i=1dmi−ni>0:∣m∣=∣n∣=q.Obviously,λmin=2, so that the lower chaos grade is g=2/q. On the other hand, the largest eigenvalue in the expansion of F2, as a sum of eigenfunctions, is given by ∣m+n∣=∣m∣+∣n∣=2q. Therefore, the upper chaos grade is given by u=2.IfF=∑i=1MHqix, where 0<q1<⋯<qd, then the lower and upper chaos grades are given, respectively, by
(28)g=2∧min1≤i,j≤Mi≠j∣qi−qj∣qM,,u=2.


### 3.3. Jacobi Operator

Forα,β>−1, we consider the d-dimensional Jacobi generator L, defined for any test function f by
(29)Lfx=∑i=1dxi1−xi∂ii2fx−∑i=1dα+βxi−α∂ifxforx∈ℝd,action on L2ℝd,⊗i=1dμidxi, where
(30)μidxi=Γα+βΓαΓβxiα−11−xiβ−110,1xi,i=1,⋯,d.Its spectrumΛ is of the form
(31)Λ=−∑i=1dλmi:m1,⋯,md∈ℕ0,where λk=kk+α+β−1. Let us set
(32)F=∑m≥0λ∣m∣=λamPmα−1,β−11−2x,λ≥2,where Pmiα,βx, mi=0,1,⋯, denotes the mith Jacobi polynomial being given by
(33)Pmiα,βx=−1mi2mimi!1−x−α1+x−βdmidxmi1−xα+mi1−xβ+mi.Recall thatpFq denotes the generalized hypergeometric function with p numerator and q denominator, given by
(34)pFqapbqx=∑k=0∞a1ka2k⋯apkb1kb2k⋯bpkxkk!,where
(35)αk=Γα+kΓα.Then the Jacobi polynomials can be expressed as(36)Pmiα,βx=α+1mimi!2F1−mi,α+β+mi+1α+11−x2.The well-known product formula of Jacobi polynomials yields that(37)F2=∑m≥0∑i=1dλmi=λ∑n≥0∑i=1dλni=λaman∑r≤m∧nCm,n,rPm+n−2rα−1,β−11−2x=∑m≥0∑i=1dλmi=λ∑n≥0∑i=1dλni=λCm,n,m∧nPm+n−2m∧nα−1,β−11−2x+⋯⋯+Cm,n,0Pm+nα−1,β−11−2x,where
(38)Cm,n,r=∫0,1dPm+n−2rα−1,β−11−2xPmα−1,β−11−2xPnα−1,β−11−2x⊗i=1dμidxi.First, observe that(39)λmi+ni=mi+nimi+ni+α+β−1=λmi+λni+2mini.It follows, from (39), that for any indices m and x such that ∑i=1dλmi=∑i=1dλxi=λ and ∣m∣=degF,
(40)maxx≥0∣x∣=degF=2λ+2maxx≥0∣x∣=degF∑i=1dmixi,where the notation degF denotes the degree of F. Successive applications of the arguments for (39) yield that
(41)λ∣m∣=∑i=1dλmi+2∑1≤i<j≤dmimj.For any another indexx such that ∣x∣=degF and λ∣x∣=λ, we have, from (41), that
(42)∑1≤i<j≤dmimj=∑1≤i<j≤dxixj,so that
(43)∑i=1dxi2=∑i=1dmi2.Letfx=∑i=1dmixi. Now, we find a point yielding the maximum value of fx under the restriction given by (43). Obviously, Lagrange’s method shows that
(44)mi−2δxi=0foralli=1,⋯,d,where δ is a Lagrange multiplier. Plugging xi=mi/2δ into (43) yields that 4δ2=1, so that xi=mi for all i=1,⋯,d. Therefore, it follows, from (40), that
(45)maxx≥0∣x∣=degF∑i=1dλmi+xi=∑i=1dλ2mi.Hence, the upper chaos grade is given by(46)u=maxm≥0∑i=1dλmi=λ∑i=1dλ2miλ.Next, we find the lower chaos grade ofF. From (37), the square of F can be expanded as a sum of eigenfunctions with the smallest eigenvalue, among positive eigenvalues, being given by
(47)λmin=minm≥0,n≥0∑i=1dλmi=∑i=1dλni=λ∑i=1dλ∣mi−ni∣:m=n=degFandmi≠niforsomei=1,⋯,d.Whenmi−ni=0 for i≠k,ℓ∈1,⋯,d and mi−ni=1 for i=k,ℓ∈1,⋯,d, the sum ∑i=1dλmi−ni has the minimum value. Hence, λmin=2λ1, so that the lower chaos grade is given by g=2α+β/λ.


## 3.1. Chaos Grade

Fix a probability spaceE,F,μ. We assume that −L has a discrete spectrum Λ=0=λ0<λ1<⋯<λk<⋯. Obviously, the zero is always an eigenfunction such that −L1=0. By the assumption on the spectrum of L, one has that
(15)L2E,F,μ=⊕ℓ=0∞KerL+λℓId.Now, we define chaotic random variables as follows.Definition 2.
Suppose thatF∈⊕i=1MKerL+λℓiId⊕KerL, where 0<λℓ1<λℓ2<⋯<λℓM and Fℓi∈KerL+λℓiId for i=1,⋯,M. The random variable F is called chaotic if there exist u>1 and g<1 such that uλℓM∈Λ and gλℓM∈Λ satisfy
(16)F2∈KerL+gλℓMId⊕⋯⊕KerL+uλℓMId⊕KerL.In this case, the largest numberg satisfying (16) is called the lower chaos grade of F. On the other hand, the smallest number u satisfying (16) is called the upper chaos grade of F.Remark 3.
(1)
The authors in [16] define the chaos grade of F, corresponding to the upper chaos grade in Definition 2, in the case where F is an eigenfunction with respect to an eigenvalue of the generator L. In this paper, we introduce the lower chaos grade of F, which will be used to obtain a better estimate for the four moments theorem than the estimate given in Theorem 3.9 of [16] in the particular case where the target distribution is a standard Gaussian distribution(2)
LetFℓk∈KerL+λℓkId for k∈1,⋯,M. If F is a chaotic random variable, then FℓiFℓj, i,j∈1,⋯,M, can be expanded as a sum of eigenfunctions with the eigenvalue of the largest magnitude λmaxℓi,ℓj and the eigenvalue of the smallest magnitude λminℓi,ℓj>0, i.e.,(17)FℓiFℓj∈KerL+λminℓi,ℓjId⊕⋯⊕KerL+λmaxℓi,ℓjId⊕KerL
From (17), it follows that
(18)F2∈KerL+λminId⊕⋯⊕KerL+λmaxId⊕KerL,where λmin=min1≤i,j≤Mλminℓi,ℓj and λmax=max1≤i,j≤Mλmaxℓi,ℓj.
(3)
In the paper [16], the authors describe how the a chaos grade, corresponding to the upper chaos grade in our works, behaves under tensorization. Let Li be a generator with invariant measure μi. Define a generator L=⊗i=1MLℓi by(19)L⊗i=1MFℓi=∑i=1MFℓ1×⋯×Fℓi−1×LℓiFℓi×Fℓi+1×⋯×FℓM
IfLiFi=λiFi, then F=⊗i=1MFℓi is an eigenfunction of L with eigenvalue λ=∑i=1Mλℓi. Suppose that Fℓi, 1≤i≤M, has the lower chaos grades gℓi. Then, the lower chaos grade g of F=⊗i=1MFℓi is given by
(20)g=∑i=1Mgℓiλℓi∑i=1Mλℓi.See Corollary 4.1 in [16] for the upper chaos grade of F.Next, we consider a finite dimensional eigenfunction and a finite sum of eigenfunctions to illustrate the concept on chaos grades.


## 3.2. Ornstein-Uhlenbeck Operator

We consider thed-dimensional Ornstein-Uhlenbeck generator L, defined for any test function f by
(21)Lfx=ΔfX−∑i=1dxi∂ifxforx∈ℝd,action on L2ℝd,⊗i=1dμidxi, where
(22)μidxi=12πe−xi2/2dxi.For a multi-indexm=m1,⋯,md, we define a d-dimensional Hermite polynomial by
(23)Hmx=∏i=1dHmixi.We write∣m∣=∑i=1dmi, m!=∏i=1dmi!, and m≤n if mi≤ni for all i=1,⋯,d. Let us set
(24)F=∑m≥0∣m∣=qamHmx,q≥2,where Hmi, mi=0,1,⋯, denotes the Hermite polynomial of order mi. Then, we have that F∈KerL+qId. By the well-known product formula of Hermite polynomials and a change of variables, we have that
(25)F2=∑m≥0m=q∑n≥0n=qaman∑r≤m∧nr!mrnrHm+n−2rx=∑m≥0m=q∑n≥0n=qCm,n,m∧nHm+n−2m∧nx+⋯⋯+Cm,n,0Hm+nx,where
(26)Cm,n,r=amanr!mrnr.Sincem+n−2m∧n=m1−n1,⋯,md−nd, F2 can be expanded as a sum of eigenfunctions with the smallest eigenvalue, among positive eigenvalues, being given by
(27)λmin=minm≥0,n≥0∑i=1dmi−ni>0:∣m∣=∣n∣=q.Obviously,λmin=2, so that the lower chaos grade is g=2/q. On the other hand, the largest eigenvalue in the expansion of F2, as a sum of eigenfunctions, is given by ∣m+n∣=∣m∣+∣n∣=2q. Therefore, the upper chaos grade is given by u=2.IfF=∑i=1MHqix, where 0<q1<⋯<qd, then the lower and upper chaos grades are given, respectively, by
(28)g=2∧min1≤i,j≤Mi≠j∣qi−qj∣qM,,u=2.


## 3.3. Jacobi Operator

Forα,β>−1, we consider the d-dimensional Jacobi generator L, defined for any test function f by
(29)Lfx=∑i=1dxi1−xi∂ii2fx−∑i=1dα+βxi−α∂ifxforx∈ℝd,action on L2ℝd,⊗i=1dμidxi, where
(30)μidxi=Γα+βΓαΓβxiα−11−xiβ−110,1xi,i=1,⋯,d.Its spectrumΛ is of the form
(31)Λ=−∑i=1dλmi:m1,⋯,md∈ℕ0,where λk=kk+α+β−1. Let us set
(32)F=∑m≥0λ∣m∣=λamPmα−1,β−11−2x,λ≥2,where Pmiα,βx, mi=0,1,⋯, denotes the mith Jacobi polynomial being given by
(33)Pmiα,βx=−1mi2mimi!1−x−α1+x−βdmidxmi1−xα+mi1−xβ+mi.Recall thatpFq denotes the generalized hypergeometric function with p numerator and q denominator, given by
(34)pFqapbqx=∑k=0∞a1ka2k⋯apkb1kb2k⋯bpkxkk!,where
(35)αk=Γα+kΓα.Then the Jacobi polynomials can be expressed as(36)Pmiα,βx=α+1mimi!2F1−mi,α+β+mi+1α+11−x2.The well-known product formula of Jacobi polynomials yields that(37)F2=∑m≥0∑i=1dλmi=λ∑n≥0∑i=1dλni=λaman∑r≤m∧nCm,n,rPm+n−2rα−1,β−11−2x=∑m≥0∑i=1dλmi=λ∑n≥0∑i=1dλni=λCm,n,m∧nPm+n−2m∧nα−1,β−11−2x+⋯⋯+Cm,n,0Pm+nα−1,β−11−2x,where
(38)Cm,n,r=∫0,1dPm+n−2rα−1,β−11−2xPmα−1,β−11−2xPnα−1,β−11−2x⊗i=1dμidxi.First, observe that(39)λmi+ni=mi+nimi+ni+α+β−1=λmi+λni+2mini.It follows, from (39), that for any indices m and x such that ∑i=1dλmi=∑i=1dλxi=λ and ∣m∣=degF,
(40)maxx≥0∣x∣=degF=2λ+2maxx≥0∣x∣=degF∑i=1dmixi,where the notation degF denotes the degree of F. Successive applications of the arguments for (39) yield that
(41)λ∣m∣=∑i=1dλmi+2∑1≤i<j≤dmimj.For any another indexx such that ∣x∣=degF and λ∣x∣=λ, we have, from (41), that
(42)∑1≤i<j≤dmimj=∑1≤i<j≤dxixj,so that
(43)∑i=1dxi2=∑i=1dmi2.Letfx=∑i=1dmixi. Now, we find a point yielding the maximum value of fx under the restriction given by (43). Obviously, Lagrange’s method shows that
(44)mi−2δxi=0foralli=1,⋯,d,where δ is a Lagrange multiplier. Plugging xi=mi/2δ into (43) yields that 4δ2=1, so that xi=mi for all i=1,⋯,d. Therefore, it follows, from (40), that
(45)maxx≥0∣x∣=degF∑i=1dλmi+xi=∑i=1dλ2mi.Hence, the upper chaos grade is given by(46)u=maxm≥0∑i=1dλmi=λ∑i=1dλ2miλ.Next, we find the lower chaos grade ofF. From (37), the square of F can be expanded as a sum of eigenfunctions with the smallest eigenvalue, among positive eigenvalues, being given by
(47)λmin=minm≥0,n≥0∑i=1dλmi=∑i=1dλni=λ∑i=1dλ∣mi−ni∣:m=n=degFandmi≠niforsomei=1,⋯,d.Whenmi−ni=0 for i≠k,ℓ∈1,⋯,d and mi−ni=1 for i=k,ℓ∈1,⋯,d, the sum ∑i=1dλmi−ni has the minimum value. Hence, λmin=2λ1, so that the lower chaos grade is given by g=2α+β/λ.


## 4. Fourth Moment Theorem

In this section, we derive an upper bound on Kolmogorov distancedKolF,Z, where F, not necessarily belonging to a fixed eigenspace, is a random variable related to Markov diffusion generator L, and Z is a standard Gaussian random variable.


### 4.1. Lemmas

We begin with stating a useful lemma, which is going to be frequently used in this section. Lemmas, which appeared in this section, are well-known in the particular case whereF is a functionals of Gaussian random fields.Lemma 4.
LetF∈DE and G∈DL. Then, we have
(48)EFG=EFEG+EΓF,−L−1G.Proof.
SinceDL⊆DE⊆L2E,μ, we have that E∣FG∣≤∥F∥2∥G∥2<∞ and ΓF,−L−1G∈L1E,μ. Hence, all expectations in equation (48) are well defined. By the integration by parts formula (12) and (14), we have that
(49)EFG=EFEG+EFG−EG=EFEG−EFL−L−1G=EFEG+EΓF,−L−1G.
This gives the desired result.Now, we extend the techniques developed in [20] in the case of a functional of Gaussian fields to a random variable belonging to L2E,μ. Let F∈DL. Define Γ0F=F and Γ1F=ΓF,−L−1F. If ΓjF, j≥1, is a well-defined element in DL, we set
(50)Γj+1F=ΓF,−L−1ΓjFforj=0,1,⋯.Similarly, letF∈DL; we define Γ0∗F=F and Γ1∗F=Γ1F. If Γj∗F∈DE for fixed j≤1,
(51)Γj+1∗F=−L−1F,Γj∗F.Lemma 5.
Suppose thatΓjF∈DL, j=0,1,2, with EF=0. Then, we have
(52)EΓ3F=16EF4−3EF22.Proof.
Observe thatΓ3F∈L1E,μ. Using the definition of Γ3 and Lemma 4, we have that
(53)EΓ3F=EFΓ2F.
The diffusion property and Lemma4 yield, from (53), that
(54)EΓ3F=EΓ12F2,−L−1Γ1F=12EF2ΓF,−L−1F−12EF22.
The carré champ operatorΓ and the integration by parts formula prove that the right-hand side of (54) can be computed as
(55)EΓ3F=14EF2LF−L−1F−FL−L−1F−−L−1FLF−12EF22=−14EΓF2,F−L−1F+14EF4+14EΓF2−L−1F,F−12EF22.
Again, using diffusion property, the first and third expectations in (55) can be represented as
(56)−14EΓF2,F−L−1F=−14EΓF2,F−L−1F−16EΓF3,−L−1F=−14EΓF2,F−L−1F−16EF4,and similarly,
(57)14EΓF2−L−1F,F=14E−L−1FΓF2,F+112EF4.
Plugging (56) and (57) into (55) yields that the equality (52) holds.Next, we investigate the relation betweenΓ3 and Γ3∗.Lemma 6.
Suppose thatΓjF∈DL, j=0,1,2, with EF=0. We have
(58)VarΓ1F=2EΓ3F−EΓ3∗F.Proof.
By the definition of the operatorΓ1∗ and the carré du champ operator Γ, we have that
(59)EΓF,−L−1F2=12EΓ1∗FLF−L−1F−FL−L−1F−−L−1FLF=12−EΓF−L−1F,Γ1∗F+EF2Γ1∗F+EΓF,Γ1∗F−L−1F.
Using the diffusion property, the first expectation in (59) can be written as
(60)−12EΓF−L−1F,Γ1∗F=−12EFΓ−L−1F,Γ1∗F−12E−L−1FΓF,Γ1∗F=−12EFΓ2∗F−12E−L−1FΓF,Γ1∗F.
Lemma4 shows that the second expectation in (59) can be computed as
(61)12EF2Γ1∗F=16EΓF3,−L−1F=16EF4.
The diffusion property shows that the third expectation in (59) can be represented as follows:
(62)−12EΓF,Γ1∗F−L−1F=12EΓ1∗FΓF,−L−1F+12E−L−1FΓF,Γ1∗F=12EΓ1∗F2+12E−L−1FΓF,Γ1∗F.
Plugging (60), (61), and (62) into (59) yields that
(63)EΓ1∗F2=−12EFΓ2∗F+16EF4+12EΓ1∗F2.
From Lemma4 and (63), it follows that
(64)EΓ1∗F2=13EF4−EΓ3∗F.
The above result (64) deduces that
(65)VarΓ1∗F=13EF4−3EF22−EΓ3∗F=2EΓ3F−EΓ3∗F.Hence, the desired result follows.


### 4.2. Fourth Moment Theorem

Let us define a set(66)CF=c∈ℝ:EΓ3∗F>cEΓ3F.Lemma 7.
For any random variableF related to a Markov diffusion generator L such that VarΓ1F>0, one has that CF≠∅.Proof.
Defineφx=EΓ3Fx−EΓ3∗F. Assumption and Lemma 6 yield that φ2>0. If EΓ3∗F≥0, then φ0≤0 and EΓ3F>0 by Lemma 6. Hence, there exists c∈0,2 such that φc≤0. If EΓ3∗F<0 and EΓ3F>0, then φ2>φ0>0, so that there exists a constant c∈−∞,0 such that φc≤0. On the other hand, if EΓ3∗F<0 and EΓ3F<0, then 0<φ2<φ0, which implies that we can find a constant c∈2,∞ such that φc≤0. Obviously, combining the above results proves that CF≠∅.Theorem 8 (fourth moment bound).
Suppose thatΓjF∈DL, j=0,1,2, with EF=0 and VarΓ1F>0. If the law of F is absolutely continuous with respect to the Lebesgue measure, there exists a constant c≠2 such that
(67)dKolF,Z≤2−cEΓ3F.Proof.
By Stein’s equation, we have that, forz∈ℝ,
(68)ℙF≤z−ℙZ≤z=Efz′F−FfzF,where fz is a solution of Stein’s equation. Since EF=0, we have that F=LL−1F. Therefore, by the integration by parts formula (12) and the derivation of Γ, the right-hand side of (68) can be written as
(69)Efz′F−FfzF=Efz′F−LL−1FfzF=Efz′F−ΓfzF,−L−1F=Efz′F1−ΓF,−L−1F≤∥fz′∥∞VarΓ1∗F.
Since∥fz∥∞≤1, we have, from Lemma 5 and Lemma 7 together with (69), that
(70)dKolF,Z≤2EΓ3F−EΓ3∗F≤2−cEΓ3F.Remark 9.
(1)
IfEΓ3F≥0, then we see, from the proof of Lemma 7, that c∈−∞,2, and hence, the value in the square root in (67) is of positive. On the other hand, if EΓ3F<0, then c∈2,∞. This means that the value in the squre root of upper bound also is of positive(2)
If2EΓ3F=EΓ3∗F, then, by (70), F is a random variable having a standard Gaussian distribution. Conversely, suppose that F is a random variable having a standard Gaussian distribution. Then, we have that −LF=F and ΓF,−L−1F=ΓF=1. Hence, 2Γ3F=Γ3∗F=0As far as we know, the following is the first result of the quantitative fourth moment theorem for a random variableF belonging to a sum of Wiener chaoses.Corollary 10.
LetF=Hpx+Hqx for p>q≥2, where Hp and Hq are Hermite polynomials of order pand q, respectively. Then, one has that
(71)dKolF,Z≤162−gEF4−3EF42,where the lower chaos grade g of F is given by
(72)g=2∧p−qp.Proof.
We compute the expectationEΓ3∗F. First, note that
(73)Γ1∗F=1pΓHpx,Hpx+1p+1qΓHpx,Hqx+1qΓHqx,Hqx.
Let us set(74)crp,q=r!prqr.
Obviously, the well-known product formula and the definition of carré du champ operator prove that fori,j=p,q,
(75)ΓHix,Hjx=∑r=0i∧jcri,jrHi+j−2rx.
Direct computations yield, from (73) and (75), together with Γc,Hix=0, that
(76)ΓHpx,Γ1∗F=1p∑r=0p−1crp,pr∑s=0p∧2p−rcsp,2p−rsH3p−2r−2sx+1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2rsH2p+q−2r−2sx+1q∑r=0q−1crq,qr∑s=0p∧2q−rcsp,2q−rsHp+2q−2r−2sx.
From the right-hand side of (76), it follows that
(77)1pEHpxΓHpx,Γ1∗F=1p2∑r=0p−1crp,prcp−rp,2p−rp−rEHpx2+1p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2rs×12s=p+q−2rEHpx2+1pq∑r=0q−1crq,qrcq−rp,2q−rq−rEHpx2.
On the other hand, sinceL−1c=0, we get
(78)ΓHpx,−L−1Γ1F=1p∑r=0p−1crp,pr2p−r∑s=0p∧2p−rcsp,2p−rsH3p−2r−2sx+1p+1q∑r=0qcrp,qrp+q−2r∑s=0p∧2p−rcsp,p+q−2rsH2p+q−2r−2sx+1q∑r=0q−1crq,qr2q−r∑s=0p∧2q−rcsp,2q−rsHp+2q−2r−2sx,so that
(79)EHpxΓHpx,−L−1Γ1F=12p∑r=0p−1crp,prcp−rp,2p−rEHpx2+121p+1q∑r=0qcrp,pr∑s=0p∧2p−rcsp,p+q−2r12s=p+q−2rEHpx2+12q∑r=0q−1crq,qrcq−rp,2q−rEHpx2.
From (77) and (79), we have that
(80)1pEHpxΓHpx,Γ1∗F−cEHpxΓHpx,−L−1Γ1F=12p2∑r=0p−1crp,pcp−rp,2p−rr2p−r−cpEHpx2+12p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2r2s−cp×12s=p+q−2rEHpx2+12pq∑r=0q−1crq,qrcq−rp,2q−r2q−r−cpEHpx2≔A1c+A2c+A3c.
Obviously, whenc=2/p, we have
(81)A1c≥2−cp12p2∑r=0p−1crp,pcp−rp,2p−rrEHpx2=0,A3c≥2−cp12pq∑r=0q−1crq,qrcq−rp,2q−rEHpx2=0.
Similarly, ifc=p−q/p, then
(82)A2c≥p−q−cp12p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2r×12s=p+q−2rEHpx2=0.
Therefore, ifc=2∧p−q/p, then Aic≥0 for i=1,2,3, so that
(83)1pEHpxΓHpx,Γ1∗F≥2∧p−qpEHpxΓHpx,−L−1Γ1F.
By a similar computation as for (77), one has that
(84)1pEHqxΓHpx,Γ1∗F=1p2∑r=0p−1crp,pr∑s=0p∧2p−rcsp,2p−rs12s=3p−q−2rEHqx2+1p1p+1q∑r=0qcrp,qrcp−rp,p+q−2rp−rEHqx2+1pq∑r=0q−1crq,qr∑s=0p∧2q−rcsp,2q−rs12s=p+q−2rEHqx2.
On the other hand,(85)EHqxΓHpx,−L−1Γ1F=12p∑r=0p−1crp,prp−r∑s=0p∧2p−rcsp,2p−rs12s=3p−q−2rEHqx2+1p+1q∑r=0qcrp,qrp+q−2rcp−rp,p+q−2rp−rEHqx2+12q∑r=0q−1crq,qrq−r∑s=0p∧2q−rcsp,2q−rs12s=p+q−2rEHqx2.
Therefore, one has, from (84) and (85), that
(86)1pEHqxΓHpx,Γ1∗F−cEHqxΓHpx,−L−1Γ1F=12p2∑r=0p−1crp,pp−rcp−rp,2p−rr∑s=0p∧2p−rcsp,2p−rs×12s=3p−q−2r2p−r−cpEHpx2+1p1p+1q∑r=0qcrp,qp+q−2rcsp,p+q−2rrp−rp+q−2r−cp×EHqx2+12pq∑r=0q−1crq,qq−rrcq−rp,2q−r2q−r−cp∑s=0p∧2q−rcsp,2q−rs×12s=p+q−2rEHqx2≔B1c+B2c+B3c.
Using the same arguments as for the case ofAic, i=1,2,3, yields that Bic≥0, i=1,2,3, for c=2∧p−q/p. This implies that
(87)1pEHqxΓHpx,Γ1∗F≥2∧p−qpEHqxΓHpx,−L−1Γ1F.
Similarly,(88)1qEHpxΓHqx,Γ1∗F≥2∧p−qpEHpxΓHqx,−L−1Γ1F,(89)1qEHqxΓHqx,Γ1∗F≥2∧p−qpEHqxΓHqx,−L−1Γ1F.
Combining the above results (83), (87), (88), and (89), we can show that
(90)EΓ3∗F≥2∧p−qpEΓ3F.
Hence, the proof of this corollary is completed.


## 4.1. Lemmas

We begin with stating a useful lemma, which is going to be frequently used in this section. Lemmas, which appeared in this section, are well-known in the particular case whereF is a functionals of Gaussian random fields.Lemma 4.
LetF∈DE and G∈DL. Then, we have
(48)EFG=EFEG+EΓF,−L−1G.Proof.
SinceDL⊆DE⊆L2E,μ, we have that E∣FG∣≤∥F∥2∥G∥2<∞ and ΓF,−L−1G∈L1E,μ. Hence, all expectations in equation (48) are well defined. By the integration by parts formula (12) and (14), we have that
(49)EFG=EFEG+EFG−EG=EFEG−EFL−L−1G=EFEG+EΓF,−L−1G.
This gives the desired result.Now, we extend the techniques developed in [20] in the case of a functional of Gaussian fields to a random variable belonging to L2E,μ. Let F∈DL. Define Γ0F=F and Γ1F=ΓF,−L−1F. If ΓjF, j≥1, is a well-defined element in DL, we set
(50)Γj+1F=ΓF,−L−1ΓjFforj=0,1,⋯.Similarly, letF∈DL; we define Γ0∗F=F and Γ1∗F=Γ1F. If Γj∗F∈DE for fixed j≤1,
(51)Γj+1∗F=−L−1F,Γj∗F.Lemma 5.
Suppose thatΓjF∈DL, j=0,1,2, with EF=0. Then, we have
(52)EΓ3F=16EF4−3EF22.Proof.
Observe thatΓ3F∈L1E,μ. Using the definition of Γ3 and Lemma 4, we have that
(53)EΓ3F=EFΓ2F.
The diffusion property and Lemma4 yield, from (53), that
(54)EΓ3F=EΓ12F2,−L−1Γ1F=12EF2ΓF,−L−1F−12EF22.
The carré champ operatorΓ and the integration by parts formula prove that the right-hand side of (54) can be computed as
(55)EΓ3F=14EF2LF−L−1F−FL−L−1F−−L−1FLF−12EF22=−14EΓF2,F−L−1F+14EF4+14EΓF2−L−1F,F−12EF22.
Again, using diffusion property, the first and third expectations in (55) can be represented as
(56)−14EΓF2,F−L−1F=−14EΓF2,F−L−1F−16EΓF3,−L−1F=−14EΓF2,F−L−1F−16EF4,and similarly,
(57)14EΓF2−L−1F,F=14E−L−1FΓF2,F+112EF4.
Plugging (56) and (57) into (55) yields that the equality (52) holds.Next, we investigate the relation betweenΓ3 and Γ3∗.Lemma 6.
Suppose thatΓjF∈DL, j=0,1,2, with EF=0. We have
(58)VarΓ1F=2EΓ3F−EΓ3∗F.Proof.
By the definition of the operatorΓ1∗ and the carré du champ operator Γ, we have that
(59)EΓF,−L−1F2=12EΓ1∗FLF−L−1F−FL−L−1F−−L−1FLF=12−EΓF−L−1F,Γ1∗F+EF2Γ1∗F+EΓF,Γ1∗F−L−1F.
Using the diffusion property, the first expectation in (59) can be written as
(60)−12EΓF−L−1F,Γ1∗F=−12EFΓ−L−1F,Γ1∗F−12E−L−1FΓF,Γ1∗F=−12EFΓ2∗F−12E−L−1FΓF,Γ1∗F.
Lemma4 shows that the second expectation in (59) can be computed as
(61)12EF2Γ1∗F=16EΓF3,−L−1F=16EF4.
The diffusion property shows that the third expectation in (59) can be represented as follows:
(62)−12EΓF,Γ1∗F−L−1F=12EΓ1∗FΓF,−L−1F+12E−L−1FΓF,Γ1∗F=12EΓ1∗F2+12E−L−1FΓF,Γ1∗F.
Plugging (60), (61), and (62) into (59) yields that
(63)EΓ1∗F2=−12EFΓ2∗F+16EF4+12EΓ1∗F2.
From Lemma4 and (63), it follows that
(64)EΓ1∗F2=13EF4−EΓ3∗F.
The above result (64) deduces that
(65)VarΓ1∗F=13EF4−3EF22−EΓ3∗F=2EΓ3F−EΓ3∗F.Hence, the desired result follows.


## 4.2. Fourth Moment Theorem

Let us define a set(66)CF=c∈ℝ:EΓ3∗F>cEΓ3F.Lemma 7.
For any random variableF related to a Markov diffusion generator L such that VarΓ1F>0, one has that CF≠∅.Proof.
Defineφx=EΓ3Fx−EΓ3∗F. Assumption and Lemma 6 yield that φ2>0. If EΓ3∗F≥0, then φ0≤0 and EΓ3F>0 by Lemma 6. Hence, there exists c∈0,2 such that φc≤0. If EΓ3∗F<0 and EΓ3F>0, then φ2>φ0>0, so that there exists a constant c∈−∞,0 such that φc≤0. On the other hand, if EΓ3∗F<0 and EΓ3F<0, then 0<φ2<φ0, which implies that we can find a constant c∈2,∞ such that φc≤0. Obviously, combining the above results proves that CF≠∅.Theorem 8 (fourth moment bound).
Suppose thatΓjF∈DL, j=0,1,2, with EF=0 and VarΓ1F>0. If the law of F is absolutely continuous with respect to the Lebesgue measure, there exists a constant c≠2 such that
(67)dKolF,Z≤2−cEΓ3F.Proof.
By Stein’s equation, we have that, forz∈ℝ,
(68)ℙF≤z−ℙZ≤z=Efz′F−FfzF,where fz is a solution of Stein’s equation. Since EF=0, we have that F=LL−1F. Therefore, by the integration by parts formula (12) and the derivation of Γ, the right-hand side of (68) can be written as
(69)Efz′F−FfzF=Efz′F−LL−1FfzF=Efz′F−ΓfzF,−L−1F=Efz′F1−ΓF,−L−1F≤∥fz′∥∞VarΓ1∗F.
Since∥fz∥∞≤1, we have, from Lemma 5 and Lemma 7 together with (69), that
(70)dKolF,Z≤2EΓ3F−EΓ3∗F≤2−cEΓ3F.Remark 9.
(1)
IfEΓ3F≥0, then we see, from the proof of Lemma 7, that c∈−∞,2, and hence, the value in the square root in (67) is of positive. On the other hand, if EΓ3F<0, then c∈2,∞. This means that the value in the squre root of upper bound also is of positive(2)
If2EΓ3F=EΓ3∗F, then, by (70), F is a random variable having a standard Gaussian distribution. Conversely, suppose that F is a random variable having a standard Gaussian distribution. Then, we have that −LF=F and ΓF,−L−1F=ΓF=1. Hence, 2Γ3F=Γ3∗F=0As far as we know, the following is the first result of the quantitative fourth moment theorem for a random variableF belonging to a sum of Wiener chaoses.Corollary 10.
LetF=Hpx+Hqx for p>q≥2, where Hp and Hq are Hermite polynomials of order pand q, respectively. Then, one has that
(71)dKolF,Z≤162−gEF4−3EF42,where the lower chaos grade g of F is given by
(72)g=2∧p−qp.Proof.
We compute the expectationEΓ3∗F. First, note that
(73)Γ1∗F=1pΓHpx,Hpx+1p+1qΓHpx,Hqx+1qΓHqx,Hqx.
Let us set(74)crp,q=r!prqr.
Obviously, the well-known product formula and the definition of carré du champ operator prove that fori,j=p,q,
(75)ΓHix,Hjx=∑r=0i∧jcri,jrHi+j−2rx.
Direct computations yield, from (73) and (75), together with Γc,Hix=0, that
(76)ΓHpx,Γ1∗F=1p∑r=0p−1crp,pr∑s=0p∧2p−rcsp,2p−rsH3p−2r−2sx+1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2rsH2p+q−2r−2sx+1q∑r=0q−1crq,qr∑s=0p∧2q−rcsp,2q−rsHp+2q−2r−2sx.
From the right-hand side of (76), it follows that
(77)1pEHpxΓHpx,Γ1∗F=1p2∑r=0p−1crp,prcp−rp,2p−rp−rEHpx2+1p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2rs×12s=p+q−2rEHpx2+1pq∑r=0q−1crq,qrcq−rp,2q−rq−rEHpx2.
On the other hand, sinceL−1c=0, we get
(78)ΓHpx,−L−1Γ1F=1p∑r=0p−1crp,pr2p−r∑s=0p∧2p−rcsp,2p−rsH3p−2r−2sx+1p+1q∑r=0qcrp,qrp+q−2r∑s=0p∧2p−rcsp,p+q−2rsH2p+q−2r−2sx+1q∑r=0q−1crq,qr2q−r∑s=0p∧2q−rcsp,2q−rsHp+2q−2r−2sx,so that
(79)EHpxΓHpx,−L−1Γ1F=12p∑r=0p−1crp,prcp−rp,2p−rEHpx2+121p+1q∑r=0qcrp,pr∑s=0p∧2p−rcsp,p+q−2r12s=p+q−2rEHpx2+12q∑r=0q−1crq,qrcq−rp,2q−rEHpx2.
From (77) and (79), we have that
(80)1pEHpxΓHpx,Γ1∗F−cEHpxΓHpx,−L−1Γ1F=12p2∑r=0p−1crp,pcp−rp,2p−rr2p−r−cpEHpx2+12p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2r2s−cp×12s=p+q−2rEHpx2+12pq∑r=0q−1crq,qrcq−rp,2q−r2q−r−cpEHpx2≔A1c+A2c+A3c.
Obviously, whenc=2/p, we have
(81)A1c≥2−cp12p2∑r=0p−1crp,pcp−rp,2p−rrEHpx2=0,A3c≥2−cp12pq∑r=0q−1crq,qrcq−rp,2q−rEHpx2=0.
Similarly, ifc=p−q/p, then
(82)A2c≥p−q−cp12p1p+1q∑r=0qcrp,qr∑s=0p∧2p−rcsp,p+q−2r×12s=p+q−2rEHpx2=0.
Therefore, ifc=2∧p−q/p, then Aic≥0 for i=1,2,3, so that
(83)1pEHpxΓHpx,Γ1∗F≥2∧p−qpEHpxΓHpx,−L−1Γ1F.
By a similar computation as for (77), one has that
(84)1pEHqxΓHpx,Γ1∗F=1p2∑r=0p−1crp,pr∑s=0p∧2p−rcsp,2p−rs12s=3p−q−2rEHqx2+1p1p+1q∑r=0qcrp,qrcp−rp,p+q−2rp−rEHqx2+1pq∑r=0q−1crq,qr∑s=0p∧2q−rcsp,2q−rs12s=p+q−2rEHqx2.
On the other hand,(85)EHqxΓHpx,−L−1Γ1F=12p∑r=0p−1crp,prp−r∑s=0p∧2p−rcsp,2p−rs12s=3p−q−2rEHqx2+1p+1q∑r=0qcrp,qrp+q−2rcp−rp,p+q−2rp−rEHqx2+12q∑r=0q−1crq,qrq−r∑s=0p∧2q−rcsp,2q−rs12s=p+q−2rEHqx2.
Therefore, one has, from (84) and (85), that
(86)1pEHqxΓHpx,Γ1∗F−cEHqxΓHpx,−L−1Γ1F=12p2∑r=0p−1crp,pp−rcp−rp,2p−rr∑s=0p∧2p−rcsp,2p−rs×12s=3p−q−2r2p−r−cpEHpx2+1p1p+1q∑r=0qcrp,qp+q−2rcsp,p+q−2rrp−rp+q−2r−cp×EHqx2+12pq∑r=0q−1crq,qq−rrcq−rp,2q−r2q−r−cp∑s=0p∧2q−rcsp,2q−rs×12s=p+q−2rEHqx2≔B1c+B2c+B3c.
Using the same arguments as for the case ofAic, i=1,2,3, yields that Bic≥0, i=1,2,3, for c=2∧p−q/p. This implies that
(87)1pEHqxΓHpx,Γ1∗F≥2∧p−qpEHqxΓHpx,−L−1Γ1F.
Similarly,(88)1qEHpxΓHqx,Γ1∗F≥2∧p−qpEHpxΓHqx,−L−1Γ1F,(89)1qEHqxΓHqx,Γ1∗F≥2∧p−qpEHqxΓHqx,−L−1Γ1F.
Combining the above results (83), (87), (88), and (89), we can show that
(90)EΓ3∗F≥2∧p−qpEΓ3F.
Hence, the proof of this corollary is completed.


## 5. Markov Chaos

In this section, as an application of Theorem8, chaotic random variables such that F∈KerL+λId will be considered.Theorem 11.
LetF be a chaotic eigenfunction of −L with respect to eigenvalue λ with EF=0 and VarΓ1F>0. Suppose that F has an upper chaos grade u and a lower chaos grade g.
(a)
Ifu≤2, one has that(91)dKolF,Z≤162−gEF4−3EF22(b)
Ifu>2 and EΓ3∗F>0, then there exist ϑ∗>1 and ϑ>1 such that(92)dKolF,Z≤162−gλϑ∗−1ϑ−1EF4−3EF22(c)
Ifu>2, EΓ3∗F<0 and EΓ3∗F>0, then there exist ϑ∗<1 and ϑ>1 such that(93)dKolF,Z≤162+uλ1−ϑ∗ϑ−1EF4−3EF22.(d)
Ifu>2, EΓ3∗F<0 and EΓ3F<0, then there exist ϑ∗<1 and ϑ<1 such that(94)dKolF,Z≤162−uλ1−ϑ∗1−ϑEF4−3EF22Proof.
We computeEΓ3∗F and EΓ3F. By the definition of Γ3∗, we have
(95)EΓ3∗F=12λEΓF2,Γ1F=12λE−LF2Γ1F−E−LF2EΓ1F=12λE−LF2Γ1F−EΓ1F.
On the other hand, by using Lemma4, we have that
(96)EΓ3F=12EΓF2,−L−1ΓF,−L−1F=12EF2Γ1F−EF2EΓ1F=12EF2Γ1F−EΓ1F.
We denote byJλℓF2 the projection of F2 on KerL+λℓId. From (16) in Definition 2, we know that F2 has a chaos decomposition of the form
(97)F2=∑i=1MJλℓiF2+J0F2,where λℓ1=gλ and λℓM=uλ. By orthogonality, it follows, from (95) and (96), that
(98)EΓ3∗F=14λ2∑i=1M∑j=1Mλℓi2λ−λℓjEJℓiF2JℓjF2=14λ2∑i=1Mλℓi2λ−λℓiEJℓiF22,(99)EΓ3F=14λ∑i=1M∑j=1M2λ−λℓjEJℓiF2JℓjF2=14λ∑i=1M2λ−λℓiEJℓiF22.
The proof ofa: since u≤2, it is obvious, from (98) and (99), that
(100)EΓ3∗F−cEΓ3F=14λ2∑i=1Mλℓi−cλ2λ−λℓiEJℓiF22≥14λλℓ1−cλ2−uVarF2.
Ifc≤λℓ1/λ=g, then we have, from (100), that EΓ3∗F≥cEΓ3F≥0. Obviously, c≤g<1. This fact implies that there exists c∈0,1 such that c∈CF. Application of Theorem 8 gives the desired result (91).
The proof ofb: first note that since u>2,
(101)u−2λ∑i=1MλℓiEJℓiF22>0,(102)u−2λ∑i=1MEJℓiF22>0.
IfEΓ3∗F>0, there exist ϑ∗>1 and ϑ>1, from (102), such that
(103)EΓ3∗F−cEΓ3F=14λ2∑i=1Mλℓiuλ−λℓiEJℓiF22−u−2λ∑i=1MλℓiEJℓiF22−cλ∑i=1Muλ−λℓiEJℓiF22+cu−2λ∑i=1MEJℓiF22=u−24λϑ∗−1∑i=1MλℓiEJℓiF22−cϑ−1∑i=1MEJℓiF22≥u−24λϑ∗−1gλ−cϑ−1EF4.
Now, we take(104)c=gλϑ∗−1ϑ−1,so that the right-hand side of (103) is equal to 0. Since 2EΓ3F>EΓ3∗F, we have that
(105)ϑ∗−1ϑ−1<2∑i=1MEJℓiF22∑i=1MλℓiEJℓiF22<2gλ.
This inequality (105) shows that c∈0,2, where c is given by (104). Hence, applying Theorem 8 yields the bound (92).
The proof ofc: since EΓ3∗F<0 and EΓ3F>0, there exist ϑ∗<1 and ϑ>1, by the similar estimate as for the case of (b), such that
(106)EΓ3∗F−cEΓ3F=u−24λϑ∗−1∑i=1MλℓiEJℓiF22−cϑ−1∑i=1MEJℓiF22≥u−24λϑ∗−1uλ−cϑ−1EF4.
Taking(107)c=uλϑ∗−1ϑ−1yields that the right-hand side of (106) is equal to 0. Since ϑ∗<1 and ϑ>1, the constant c given in (107) belongs to −∞,0. Application of Theorem 8 proves the bound (93).
The proof ofd: since EΓ3∗F<0 and EΓ3F<0, there exist ϑ∗<1 and ϑ<1 such that
(108)EΓ3∗F−cEΓ3F≥u−24λϑ∗−1uλ−cϑ−1EF4.
The right-hand side of (108) is equal to 0 when one takes
(109)c=uλϑ∗−1ϑ−1.
Note that(110)1−ϑ∗1−ϑ>2∑i=1MEJℓiF22∑i=1MλiEJℓiF22>2uλ.
This estimate (110) shows that c given in (109) belongs to 2,∞ from which we obtain the bound (93).Remark 12.
Suppose that the target distributionν in Theorem 4.9 of [16] is a standard Gaussian measure. If a chaotic random variable Fn, with EFn=0 and EFn=1, satisfies supnun>2, then the bound (3.7) in [16] becomes
(111)dKolFn,Z≤231−un4EFn4−3+un−22VarFn2.Even when the fourth moment ofFn in the first term of (111) converges to 3, the sequence Fn does not converge, in distribution sense, to a standard Gaussian random variable Z, due to the second term in (111). It means that the fourth moment theorem of Theorem 1 is not working. The bounds in (b), (c), and (d) of Theorem 11 show that the fourth moment theorem holds by removing the second term in (111) even if an upper chaos grade is strictly greater than 2.Remark 13.
The Ornstein-Uhlenbeck operatorL on E=ℝd is then given by
(112)Lf=Δf−x·∇f.
The carré du champ operatorΓ is the usual gradient operator Γf,f=∇f2. In infinite-dimensional setting, the infinite-dimensional Ornstein-Uhlenbeck generator on Wiener space can be obtained with Wiener measure as an invariant distribution. If F is an element belonging to a fixed Wiener chaos with order q, i.e, F=Iqf, f∈H⊗q, the well-known product formula of multiple stochastic integrals gives that
(113)F2=∑r=0qr!qr2I2q−rf⊗rf,where f⊗rf is the contraction of the kernel f. This expansion of the square of F shows, from the definition of the chaos grade, that u=2 and g=2/q. In this case λq=qandg=2/q, it follows, from the above bound (91), that
(114)dKolF,Z≤q−13qEF4−3EF22.
In this specific case, the above bound (114) has been applied to obtain the Berry–Esséen bound for parameter estimation of fractional Ornstein-Uhlenbeck processes (see [21]). Furthermore, the authors of [20] derive a bound for the form F/G to obtain an optimal Berry-Esséen bound for parameter estimation of stochastic partial differential equation.


## 6. Conclusions

In this paper, we derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator. Significant features of our work are that (a) it provides a better estimate than the previous one; (b) it contains square integrable random variables unlike all previous works in this research line, where the authors deal with only random variables to belong to a fixed eigenspace (or Wiener chaos); and (c) unlike the previous result, the fourth moment theorem holds even if the upper chaos grade is strictly greater than 2. Future research plans will study whether our methods are applicable in other (non-Gaussian) target measures (for example, a bound for gamma or beta approximation).



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