Weak convergence on Wiener space: targeting the first two chaoses

TL;DR

This paper establishes necessary and sufficient conditions for the convergence in law of sequences of random variables in the sum of the first two Wiener chaoses, using Malliavin calculus and Gamma-operators, extending prior results.

Contribution

It provides a complete characterization of convergence in law for sequences in the first two Wiener chaoses, including multiple Wiener integrals, and explores stable convergence and limitations on target variables.

Findings

Conditions for convergence in law are characterized precisely.

Results extend previous work by Azmoodeh, Peccati, and Poly (2014).

Certain classes of target variables are shown to be unattainable.

Abstract

We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos.

Equations

$$X=I_1(f_1)+I_2(f_2)=aN+\sum _{i=1}^{k_1}{b}_i(R_i^2-1)+\sum _{i=1}^{k_2}[c_i(P_i^2-1)+d_i{P}_i],$$

$$X=a{U}_0+\sum _{i=1}^n{\lambda }_i(U_i^2-1),$$

$$X=I_1(f_1)+I_2(f_2)=aN+\sum _{i=1}^{k_1}{b}_i(R_i^2-1)+\sum _{i=1}^{k_2}[c_i(P_i^2-1)+d_i{P}_i],$$

$$P(x)=x^{1+1_{[a\ne 0]}}\prod _{j=1}^{k_1}(x-b_j)\prod _{j=1}^{k_2}(x-c_j{)}^2.$$

$$X_3\stackrel{\text{Law}}{=}\sum _{i=1}^k{\alpha }_i(N_i^2-1),N_1,\dots ,N_k\stackrel{\text{i.i.d.}}{\sim }\mathcal{N}(0,1),$$

$${\mathcal{M}}_{\eta }:=\frac{-1-\eta }{2\sqrt{1+{\eta }^2}}(X_1^2-1)+\frac{-1+\eta }{2\sqrt{1+{\eta }^2}}(X_2^2-1),\eta \in [0,1],$$

$$d_{\mathcal{H}}(F,G)=\underset{h\in \mathcal{H}}{\sup }|E[h(F)]-E[h(G)]|,$$

$$X_3=\sum _{i=1}^k{\alpha }_i(N_i^2-1),$$

$$d_{W_2}(U,V):={\left(\inf E\left[\Vert X-Y{\Vert }_d^2\right]\right)}^{\frac{1}{2}},$$

$$d_{W_2}(F_n,X_3)\le C\left(\sqrt{\Delta (F_n)}+\sum _{r=2}^{k+2}|{\kappa }_r(F_n)-{\kappa }_r(X_3)|\right),$$

$$d_{W_2}(F_n,X_3)\le C\sqrt{\Delta (F_n)},$$

$$d_{W_2}(F_n,X_3)\ge C^{\prime }\sqrt{\Delta (F_n)},$$

$$d_{W_2}(F_n,X_3)\le C{\Delta }_n|\log ({\Delta }_n){|}^{\Theta },$$

$$X_4=\sum _{i=1}^k\left[{\alpha }_i(N_i^2-1)+{\beta }_i{N}_i\right],N_1,\dots ,N_k\stackrel{i.i.d.}{\sim }\mathcal{N}(0,1).$$

$$W(h)=I_1(h)={\int }_0^{T}h(t)d{W}_t,$$

$$E[W(h)W(g)]=E[I_1(h)I_1(g)]=⟨h,g{⟩}_H.$$

$$I_p(f_p)$$

$$:=p!{\int }_0^T\left({\int }_0^{t_p}\dots {\int }_0^{t_3}\left({\int }_0^{t_2}{f}_p(t_1,\dots ,t_p)d{W}_{t_1}\right)d{W}_{t_2}\dots d{W}_{t_{p-1}}\right)d{W}_{t_p}.$$

$$E[I_p(f_p)I_q(g_q)]=1_{[p=q]}p!⟨f_p,g_q{⟩}_{H^{\otimes p}}=1_{[p=q]}{\int }_{[0,T{]}^p}{f}_p{g}_{q}d{\lambda }_T^p.$$

$$(f_p{\otimes }_r{g}_q)(t_1,\dots ,t_{p+q-2r})$$

$$I_p(f_p)I_q(g_q)=\sum _{r=0}^{p\wedge q}r!\left(\begin{array}{c}p\\ r\end{array}\right)\left(\begin{array}{c}q\\ r\end{array}\right)I_{p+q-2r}(f_p{\tilde{\otimes }}_r{g}_q).$$

$$DF:=$$

$$=\sum _{i=1}^k{h}_i(t){\partial }_{i}f\left({\int }_0^T{h}_i(s)d{W}_s,\dots ,{\int }_0^T{h}_k(s)d{W}_s\right).$$

$$E\left[⟨DF,v{⟩}_H\right]\le c_v\sqrt{E[F^2]},$$

$$\Vert F{\Vert }_{1,2}:={\left(E\left[F^2\right]+E\left[\Vert DF{\Vert }_H^2\right]\right)}^{1/2}.$$

$$E[F\delta (v)]=E[⟨v,DF{⟩}_H].$$

$$\delta (Fv)=F\delta (v)-⟨DF,v{⟩}_H.$$

$$D_{x_1}{D}_{x_2}\dots D_{x_l}{\int }_0^T\dots {\int }_0^{T}f(y_1,\dots ,y_k)d{W}_{y_1}\dots d{W}_{y_k}$$

$$=\frac{k!}{(k-l)!}{\int }_0^T\dots {\int }_0^{T}f(y_1,\dots ,y_{k-l},x_1,\dots ,x_l)d{W}_{y_1}\dots d{W}_{y_{k-l}}.$$

$$E\left[F{\delta }^l(v)\right]=E[⟨D^{l}F,v{⟩}_{H^{\otimes l}}],$$

$$\Vert F{\Vert }_{m,p}={\left(E[|F{|}^p]+\sum _{i=1}^{m}E\left[\Vert D^{i}F{\Vert }_{H^{\otimes i}}^p\right]\right)}^{1/p}$$

$${\Gamma }_i(F)=⟨DF,-D{L}^{-1}{\Gamma }_{i-1}(F){⟩}_H,\text{for }i\ge 1,$$

Full text

60F05; 60G15; 60H07,

Brownian motion, Malliavin calculus, multiple Wiener integrals, weak convergence, limit theorems, cumulants, stable convergence,

1. 1 Introduction

2. 1.1 Overview

3. 1.2 History and motivation

4. 1.3 Results and plan

5. 2 Preliminaries

6. 2.1 Multiple Wiener integrals

7. 2.2 Malliavin calculus

8. 2.3 Cumulants and $\Gamma$-operators

9. 2.4 Stable convergence

10. 3 Main results

11. 4 Applications

12. 4.1 Recovering classical criteria

13. 4.2 New applications

14. 5 Stable convergence

15. 6 Appendix

1 Introduction

1.1 Overview

The aim of this paper is to provide new criteria for non-central convergence in law for sequences of polynomial functionals of a Brownian motion $W$. In particular, we consider the convergence in law of a sequence of random variables $\left\{F_{n}\right\}_{n}$ to a target variable $X$, where:

• •

the random variables $F_{n}$ have a representation of the form $F_{n}=\sum_{p=1}^{m}I_{p}(f_{n,p})$ for a fixed $m\geq 2$, where $I_{p}(\cdot)$ is the Wiener integral of order $p$ with respect to the Brownian motion $W$;

• •

the target variable $X$ lives in the sum of the first two Wiener chaoses associated with $W$ and can be represented as:

[TABLE]

where all coefficients $b_{i}$, $c_{j}$ and $d_{j}$ are non-zero and $N,R_{i},P_{j}$ are independent standard normal variables for $1\leq i\leq k_{1}$ and $1\leq j\leq k_{2}$. We shall see that this representation covers in particular random variables of the form:

[TABLE]

where all coefficients $\lambda_{i}$ are non-zero, $U_{1},\ldots,U_{n}$ are independent standard normal variables and $U_{0}$ is a standard normal variable which may be correlated to $U_{1},\ldots,U_{n}$.

Our main result (Theorem 1.1) gives a necessary and sufficient criterion for the convergence in law to $X$:

Theorem 1.1**.**

Consider $0\leq k_{1},k_{2}<\infty$ and

[TABLE]

where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Suppose that at least one of the parameters $a,k_{1},k_{2}$ is non-zero. Consider a sequence $\left\{F_{n}\right\}_{n}$ of non-zero random variables such that $F_{n}=\sum_{i=1}^{p}I_{i}(f_{n,i})$ for $p\geq 2$ fixed and $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq p$. Define:

[TABLE]

As $n\to\infty$, the following conditions (a) and (b) are equivalent:

1. (a)

2. (1)

$\displaystyle{\kappa_{r}(F_{n})\to\kappa_{r}(X),\quad\textrm{ for }r=1,\ldots,\textrm{deg}(P)}$, 2. (2)

\displaystyle{\operatorname{\mathbb{E}}\left[\left|\operatorname{\mathbb{E}}\left[\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]\right)\Big{|}F_{n}\right]\right|\right]\to 0}, 2. (b)

$F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$.

In this paper, we consider functionals of a Brownian motion. By a standard isometry argument (see e.g. [21, Section 2.2]), the results immediately extend to the framework of an isonormal Gaussian process on a general real separable Hilbert space $H$.

The notations used in this theorem are introduced in Section 2. In particular, the representation of $X$ in Eq. (1.1) is detailed in Eq. (3.3) and (3.5), $\kappa_{r}(Y)$ is the $r$-th cumulant of a random variable $Y$ and the sequence $\left\{\Gamma_{i}(F_{n})\right\}_{i}$ is defined recursively using Malliavin operators. In Section 3 the representation of the target random variable in Eq. (1.1) is derived.

Our main result unifies, generalizes and extends previous findings. More precisely, Theorem 1.1:

• •

further extends to a non-central setting the seminal paper [21] and in particular [15, Theorem 5.3.1] by dealing with central limit theorems for sequences living in a finite sum of Wiener chaoses,

• •

extends [16, Theorem 3.4] by considering a sequence $\left\{F_{n}\right\}_{n}$ of random variables which are no longer restricted to the second Wiener chaos but live in a finite sum of Wiener chaoses,

• •

extends [13, Theorem 1.2], [4, Theorem 3.2] and [6, Proposition 1.7] by considering target variables involving linear combinations of independent $\chi^{2}$ distributed random variables and adding a possibly correlated normal variable,

• •

improves [4, Theorem 3.2] by replacing $L^{2}$-convergence with $L^{1}$-convergence and finding thus a necessary and sufficient criterion for convergence in law for a large class of target variables, in particular for linear combination of independent central $\chi^{2}$ distributed target variables.

In addition to these applications which are discussed in Section 4, Theorem 1.1 is used to investigate stable convergence in Section 5.

1.2 History and motivation

The study of convergence in law for sequences of multiple Wiener integrals, by variational techniques, has been the object of an intense study in recent years. The starting point of this line of research is [21]. In this reference and later in [20], the authors gave necessary and sufficient criteria for the convergence in law of a sequence of multiple Wiener integrals $I_{p}(f_{n,p})$ to a standard normal variable $N$: If the functions $f_{n,p}$ are symmetric in the $p\geq 2$ variables with $\operatorname{\mathbb{E}}[I_{p}(f_{n,p})^{2}]\to 1$, as $n\to\infty$, then the following conditions are equivalent:

1. (i)

$\operatorname{\mathbb{E}}[I_{p}(f_{n,p})^{4}]\to 3$, as $n\to\infty$, 2. (ii)

$\|f_{n,p}\otimes_{l}f_{n,p}\|_{H^{\otimes(2p-2l)}}\to 0$, as $n\to\infty$, for every $l=1,\ldots,p-1$, 3. (iii)

$\|DI_{p}(f_{n,p})\|_{H}^{2}\stackrel{{\scriptstyle L^{2}(\Omega)}}{{\to}}p$, as $n\to\infty$, 4. (iv)

$I_{p}(f_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}N$, as $n\to\infty$.

Notice that $D$ is the standard Malliavin derivative operator and $f_{n,p}\otimes_{l}f_{n,p}$ is the contraction of order $l$, see Section 2.2. The equivalence of (i), (ii) and (iv) has been found in [21], the equivalence of either one of these conditions with (iii) has been proved later in [20]. Considering the proof of [20, Theorem 4], it is easy to see that condition (iii) above can be replaced by:

1. (iii’)

$\operatorname{\mathbb{E}}\left[\|DI_{p}(f_{n,p})\|_{H}^{2}\,|\,I_{p}(f_{n,p})\right]\stackrel{{\scriptstyle L^{1}(\Omega)}}{{\to}}p$, as $n\to\infty$.

This is remarkable since conditions of this form play a crucial role in [4] and in the main result of the present paper. Since this characterisation has been published, limit theorems have been extended beyond standard normal target variables. In [13], the authors establish, for $p\geq 2$ even, necessary and sufficient conditions for a sequence $\left\{I_{p}(f_{n,p})\right\}_{n}$ of multiple Wiener integrals to converge in law to a Gamma random variable. The conditions found by Nourdin and Peccati use contractions, convergence of the first moments of $I_{p}(f_{n,p})$ and Malliavin derivatives. In particular, the results of [13] cover the convergence in law of a sequence of multiple Wiener integrals to a random variable $X_{1}\stackrel{{\scriptstyle\textrm{Law}}}{{=}}\sum_{i=1}^{k}(N_{i}^{2}-1)$ with $N_{i}$ independent standard normal variables and $1\leq k<\infty$. $X_{1}$ has a centered $\chi^{2}$ law with $k$ degrees of freedom. The authors also prove that the convergence is stable in this case. More results about stable convergence can be found in [22] and in the more recent work [11].

For the case $p=2$, linear combinations of independent centered $\chi^{2}$ distributed random variables are important since it is known that every element $X_{2}$ of the second Wiener chaos has a representation of the form $X_{2}\stackrel{{\scriptstyle\textrm{Law}}}{{=}}\sum_{i=1}^{n}\alpha_{i}(N_{i}^{2}-1)$, where $1\leq n\leq\infty$ and $\left\{N_{i}\right\}_{i}$ is a sequence of independent standard normal variables, see [9, Theorem 6.1]. It is proved in [16], that every sequence $\left\{I_{2}(f_{n,p})\right\}_{n}$ which converges in law has a limit of the form $\alpha_{0}N_{0}+\sum_{i=1}^{n}\alpha_{i}(N_{i}^{2}-1)$, where $N_{i}$ are independent standard normal variables. In [16, Theorem 3.4] the authors use cumulants and a polynomial $Q$ to characterise this convergence in law if $n<\infty$.

The idea of using polynomials to find necessary and sufficient conditions for the convergence in law has proved to be useful. In [4], the authors consider the more general problem of finding necessary and sufficient conditions for a sequence $\left\{F_{n}\right\}_{n}$ to converge to a random variable with a representation of the form

[TABLE]

where $F_{n}$ are random variables living in a (fixed) finite sum of Wiener chaoses, see Section 2.1. For $k<\infty$, their main finding, [4, Theorem 3.2] provides a necessary and a sufficient condition for $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X_{3}$, as $n\to\infty$, in terms of Malliavin operators $\Gamma_{i}$, defined in Section 2.3.

Linear combinations of independent centered $\chi^{2}$ distributed random variables as in Eq. (1.2) are of great interest because of their role within the second Wiener chaos. This class of random variables is important in stochastic geometry as well. In [10], the authors consider the two-dimensional torus and prove the weak convergence of the normalized nodal length of the so-called ‘arithmetic random waves’, to a target variable $\mathcal{M}_{\eta}$ defined by:

[TABLE]

where $X_{1},X_{2}$ are independent standard normal variables. An important element of the proof is the fact that the Wiener chaos expansion of the normalised nodal length is dominated by its fourth order chaos component.

Another line of research, which is closely connected to the previous results, investigates the convergence of sequences living in a finite sum of Wiener chaoses by using distances between probability measures. In this context, the distance between two laws $F$ and $G$ is defined as:

[TABLE]

where $\mathscr{H}$ is a class of functions. Different classes $\mathscr{H}$ lead to distances such as the Wasserstein, total variation and Kolmogorov distance, see [15, Appendix C] for details. Upper bounds for the total variation and smooth distances are proved in [14] and [17]. For most of these results, the distance of a distribution to a centered normal distribution or a centered $\chi^{2}$ distribution with $\nu$ degrees of freedom is considered. Recently, in [6] a new estimate is proved for the Wasserstein distance of a distribution to a centered $\chi^{2}$ distribution with $\nu$ degrees of freedom. In particular, the authors find a new necessary and sufficient criterion for sequences living in a fixed Wiener chaos to converge in law to $X_{1}$ defined above.

In [1] the authors consider target variables of the form given in Eq. (1.2):

[TABLE]

where the coefficients are not necessarily pairwise distinct and $N_{1},\ldots,N_{k}$ are independent standard normal variables. The authors discuss Stein’s method for this class of target variables and apply a new and original Fourier-based approach to derive a Stein-type characterisation. The polynomials used in [16, 4] and $\Gamma$-operators, see [4, 15], are combined with the integration by parts formula of Malliavin calculus to derive a Stein operator which allows to characterise target variables as in Eq. (1.2). The authors consider a linear combination of $\Gamma$-operators which shall be generalized in the present paper and the 2-Wasserstein distance. In general, the 2-Wasserstein distance between the laws of random vectors $U$ and $V$ is defined as follows:

[TABLE]

where the infimum is taken over all joint distributions of $X$ and $Y$ with respective marginals $U$ and $V$, and $\|\cdot\|_{d}$ stands for the Euclidean norm on $\mathbb{R}^{d}$, see [2, Definition 1.1]. It is shown that:

[TABLE]

where $C$ is independent of $n$, the quantity $\Delta(F_{n})$ can be expressed in terms of cumulants $\kappa_{r}$ and polynomials, the sequence $\left\{F_{n}\right\}_{n}$ must satisfy several conditions which hold in particular for sequences living in the second Wiener chaos. It is proved in particular that:

[TABLE]

if $\dim_{\mathbb{Q}}\textrm{span }\left(\alpha_{1}^{2},\ldots,\alpha_{k}^{2}\right)=k$, see Eq. (1.2). This shows that $\Delta(F_{n})\to 0$, as $n\to\infty$, is sufficient for convergence in the 2-Wasserstein metric which implies convergence in law. In [2], it is shown that the convergence of the cumulants can not be omitted in the general case. In addition to the upper bound found in [1], a lower bound for the 2-Wasserstein metric is derived in [2], namely:

[TABLE]

where $C^{\prime}>0$ is independent of $n$ and the sequence $\left\{F_{n}\right\}_{n}$ lives in the second Wiener chaos. In other words, for sequences living in the second Wiener chaos, weak convergence to $X_{3}$ is equivalent to $\Delta(F_{n})\to 0$, as $n\to\infty$, if $\dim_{\mathbb{Q}}\textrm{span }\left(\alpha_{1}^{2},\ldots,\alpha_{k}^{2}\right)=k$. The so-called Stein-Tikhomirov method is considered in [3]. This method can be seen as a combination of Stein’s method with other methods to measure the rate of convergence of a sequence of random variables. The authors consider in particular the Stein-type characterisation found in [1, Theorem 2.1] and apply their version of the Stein-Tikhomirov method. The same linear combination of $\Gamma$-operators as in [1] is used and the so-called transfer-principle allows to find upper bounds on smooth Wasserstein distances using upper bounds on the difference of the characteristic functions of the approximating sequence and the target variable. For sequences living in the sum of the first $p$ Wiener chaoses and constants $C,\Theta>0$ depending only on $p$, the following bound is proved:

[TABLE]

where $\Delta_{n}$ is expressed in terms of cumulants and $\Gamma$-operators. In particular, if $k\geq 3$ in Eq. (1.2), we have for the Kolmogorov distance that $d_{\textrm{Kol}}(F_{n},X_{3})\leq B\,\sqrt{\Delta_{n}}$. Finally the authors find bounds for $\Delta_{n}$ if $\left\{F_{n}\right\}_{n}$ lives inside a fixed Wiener chaos and $k=1$ in Eq. (1.2). Even though the aforementioned papers present important new results for target variables living in the second Wiener chaos, none of them considers target variables living in the sum of the first two Wiener chaoses with possibly correlated first and second order components.

The results of [4, Theorem 3.2] are the starting point of the present work. As anticipated, we shall consider a sequence $\left\{F_{n}\right\}_{n}$ of random variables living in a finite (fixed) sum of Wiener chaoses and provide necessary and sufficient conditions for $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X_{4}$, as $n\to\infty$, where $k<\infty$ and $X$ has the following, more general form:

[TABLE]

The representation in Eq. (1.4) is equivalent to one in Eq. (1.1), where we have dropped all vanishing coefficients and regrouped independent normal variables. Both representations (1.1) and (1.4) are useful for the discussion to follow. Random variables as in Eq. (1.4) are important since every random variable living in the sum of the first two Wiener chaoses has a representation of this form, with $k\leq\infty$, see [9, Theorem 6.2]. Our conditions make, as in [4], use of the operators $\Gamma_{i}$. Clearly such a result can be seen as extension of [4, Theorem 3.2]. For sequences of random variables living in a finite sum of Wiener chaoses, we shall apply our methods and results to derive necessary and sufficient criteria to prove stable convergence to target variables with representations as in Eq. (1.4)

1.3 Results and plan

The paper is organized as follows:

In Section 2, we introduce the necessary notations and give a brief introduction to Malliavin calculus. The basic elements of this theory shall be needed in the forthcoming proofs. 2. -

In Section 3, we prove our characterisation in Theorems 3.8 and 3.11. The main Theorem 1.1 is then a direct consequence of these theorems. 3. -

In Section 4, we apply Theorem 1.1 to several situations, such as the convergence in law to a normal variable or a centered $\chi^{2}$ distributed random variable with $k_{1}$ degrees of freedom. We shall also recover the results of [4]. In Theorem 4.6, we give sufficient conditions, based only on cumulants and contractions. We conclude this section by giving a criterion which excludes certain classes of target variables for sequences living in a chaos of odd order. 4. -

In Section 5, we give criteria which can be used to determine whether a sequence converges stably.

2 Preliminaries

2.1 Multiple Wiener integrals

The reader is referred to [15], [19] or [5] for a detailed introduction to multiple Wiener integrals. Consider the real Lebesgue space $H=L^{2}([0,T],\lambda_{T})$, where $\lambda_{T}$ is the Lebesgue measure on $[0,T]$. The real separable Hilbert space $H$ is endowed with the standard scalar product $\langle h,g\rangle_{H}:=\int_{0}^{T}hgd\lambda$ for all $h,g\in H$. We write $H^{\otimes p}$ for $L^{2}([0,T]^{p},\lambda_{T}^{p})$, where $\lambda_{T}^{p}:=\lambda^{p}|[0,T]^{p}$, and define $H^{\odot p}$ as the subspace of $H^{\otimes p}$ containing exactly the functions which are symmetric on a set of Lebesgue measure $T^{p}$. Consider a complete probability space $\left(\Omega,\operatorname{\mathbb{P}},\mathcal{F}\right)$ and a standard Brownian motion $(W_{t})_{t\in[0,T]}$ with respect to $\operatorname{\mathbb{P}}$ and the filtration $(\mathcal{F}_{t})_{t\in[0,T]}$. Define for every $h\in H$:

[TABLE]

then $W(h)\in L^{2}(\Omega):=L^{2}\left(\Omega,\operatorname{\mathbb{P}}\right)$, in other words $W(h)$ is square-integrable. We have for $h,g\in H$:

[TABLE]

More generally the $q$-th Wiener chaos is defined as closed linear subspace of $L^{2}(\Omega)$ which is generated by the random variables of the form $H_{q}(W(h))$ where $h\in H$ with $\|h\|_{H}=1$ and $H_{q}$ is the $q$-th Hermite polynomial. The elements of the $q$-th Wiener chaos can be represented as multiple Wiener integrals. For every $f_{p}\in H^{\odot p}$:

[TABLE]

It is well known that every $F\in L^{2}(\Omega)$ has a representation of the form $F=\sum_{p=0}^{\infty}I_{p}(f_{p})$, where $I_{0}(f_{0})=f_{0}=\operatorname{\mathbb{E}}[F]$ and the right-hand side converges in $L^{2}(\Omega)$. We have moreover for $p,q\geq 1$:

[TABLE]

For two function $f_{p}\in H^{\odot p}$ and $g_{q}\in H^{\odot q}$, the contraction of $r$ indices is defined for $1\leq r\leq p\wedge q$ by:

[TABLE]

We have $f_{p}\otimes_{r}g_{q}\in H^{p+q-2r}$. The symmetrization of $f_{p}\otimes_{r}g_{q}$ is $f_{p}\tilde{\otimes}_{r}g_{q}$. We shall also need the multiplication formula for multiple Wiener integrals. For $f_{p}\in H^{\odot p}$ and $g_{q}\in H^{\odot q}$, we have:

[TABLE]

2.2 Malliavin calculus

The reader is referred to [15], [19] or [4] for a detailed introduction to Malliavin calculus. Let $k\geq 1$ and $F$ be a random variable with $F=f\left(W(h_{1}),\ldots,W(h_{k})\right)$ where $f$ is an infinitely differentiable rapidly decreasing function on $\mathbb{R}^{k}$ and $h_{1},\ldots,h_{k}\in H$. Then $F$ is called a smooth random variable and $\mathcal{S}$ is the set containing exactly the smooth random variables. The Malliavin derivative of $F$ is defined by:

[TABLE]

We have that ${D}_{t}$ is closable from $L^{2}(\Omega,\operatorname{\mathbb{P}})$ to $L^{2}(\Omega\times[0,T],\operatorname{\mathbb{P}}\otimes\lambda_{T})$, that is (see [5] or [19, Proposition 1.2.1]): If a sequence $\left\{H_{n}\right\}_{n\in\mathbb{N}}\subset L^{2}\left(\Omega,\operatorname{\mathbb{P}}\right)$ converges to [math], that is $\operatorname{\mathbb{E}}\left[H_{n}^{2}\right]\to 0$ as $n\to\infty$, and ${D}_{t}H_{n}$ converges in $L^{2}(\Omega\times[0,T],\operatorname{\mathbb{P}}\otimes\lambda_{T})$ as $n\to\infty$, then $\lim_{n\to\infty}{D}_{t}H_{n}=0$. We write Dom ${D}$ for the closed domain of ${D}$. Moreover the Malliavin derivative has a closable adjoint $\delta$ (under $\operatorname{\mathbb{P}}$). The operator $\delta$ is called the divergence operator or, in the white noise case, the Skorohod integral. The domain of $\delta$ is denoted by Dom $\delta$, it is the set of square-integrable random variables $v\in L^{2}(\Omega\times[0,T],\operatorname{\mathbb{P}}\otimes\lambda_{T})$ with:

[TABLE]

for a constant $c_{v}$ (depending on $v$) and all $F\in\mathbb{D}^{1,2}$ where $\mathbb{D}^{1,2}$ is the closure of the class of smooth random variables with respect to the norm:

[TABLE]

With the scalar product $\langle F,G\rangle_{1,2}=\operatorname{\mathbb{E}}[FG]+\operatorname{\mathbb{E}}[\langle{D}F,{D}G\rangle_{H}],$ $\mathbb{D}^{1,2}$ is a Hilbert space. If $v\in$ Dom $\delta$, then $\delta(v)$ is the element of $L^{2}(\Omega,\operatorname{\mathbb{P}})$ characterised by

[TABLE]

This relation is often called the integration by parts formula. We have the more general rule (see [15, Proposition 2.5.4]) for $F\in\mathbb{D}^{1,2}$, $v\in$ Dom $\delta$ such that $Fv\in$ Dom $\delta$:

[TABLE]

For multiple Wiener integrals and $f\in H^{\odot k}$, we have (see for instance [19, p.35]):

[TABLE]

Formula (2.3) can be generalized for the multiple divergence (see [15, p.33]): If $v\in$ Dom $\delta^{l}$ and $F\in\mathbb{D}^{l,2}$:

[TABLE]

see [15] or [19] for details. For $m\geq 1$, $p\geq 1$ and the $m$-th Malliavin derivative $D^{m}F$, we can define $\mathbb{D}^{m,p}$ as the closure of the class of smooth random variables with respect to the norm $\|\cdot\|_{m,p}$ defined by:

[TABLE]

We define $\mathbb{D}^{\infty}:=\cap_{m=1}^{\infty}\cap_{p=1}^{\infty}\mathbb{D}^{m,p}$.

2.3 Cumulants and $\Gamma$-operators

The reader is referred to [4] or [15] for a detailed introduction to cumulants, Malliavin operators and $\Gamma$-operators in particular. The $r$-th cumulant of a random variable $F$ exists if the $r$-th moment of $F$ exists and is defined as $\kappa_{r}(F):=(-i)^{r}\frac{d^{r}}{dt^{r}}\log\operatorname{\mathbb{E}}[\exp(itF)]_{|t=0}$. The operator $L$, defined as $L=-\sum_{q=0}^{\infty}qJ_{q}$, is the infinitesimal generator of the Ornstein-Uhlenbeck semi-group where $J_{q}$ is the orthogonal projection operator on the $q$-th Wiener chaos. The domain of $L$ is $\mathbb{D}^{2,2}$. $L$ admits a pseudo-inverse $L^{-1}$ and for any $F\in L^{2}(\Omega)$, we have $L^{-1}F=-\sum_{q=1}^{\infty}\frac{1}{q}J_{q}(F)$. For $F\in\mathbb{D}^{\infty}$, the sequence of random variables $\left\{\Gamma_{i}(F)\right\}_{i}\subset\mathbb{D}^{\infty}$ is recursively defined as follows:

[TABLE]

and $\Gamma_{0}(F)=F$. For $F\in\mathbb{D}^{\infty}$ and $r\geq 0$, we define: $M_{r}(F)=\Gamma_{r}(F)-\operatorname{\mathbb{E}}[\Gamma_{r}(F)]$.

2.4 Stable convergence

The concept of stable convergence is used in Section 5. The reader can find an extensive discussion of this topic in [8] or [7], the basic facts are resumed in [11]. Consider a sequence $\left\{F_{n}\right\}_{n}$ of real random variables on the complete probability space $\left(\Omega,\mathcal{F},\operatorname{\mathbb{P}}\right)$, see Section 2.1. Let $F$ be a real random variable defined on some extended probability space $\left(\Omega^{\prime},\mathcal{F}^{\prime},\operatorname{\mathbb{P}}^{\prime}\right)$. We say that $F_{n}$ converges stably to $F$, written $F_{n}\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}F$, as $n\to\infty$, if:

[TABLE]

for every $\lambda\in\mathbb{R}$ and every bounded $\mathcal{F}$-measurable random variable $Z$. Obviously, stable convergence implies convergence in law, whereas the converse does not hold in general. We notice that the $\operatorname{\mathbb{P}}$-completion of the $\sigma$-field generated by the set $\left\{I_{1}(f):f\in H\textrm{ with }\|f\|_{H}=1\right\}$ is $\mathcal{F}$. We have thus the following useful characterisation of stable convergence:

$F_{n}\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}F$ if and only if $\left(F_{n},I_{1}(f)\right)^{\top}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}\left(F,I_{1}(f)\right)^{\top}$ for every $f\in H$ with $\|f\|_{H}=1$.

3 Main results

We start with an example in order to motivate the reader.

Example 3.1*.*

Consider sequences $\left\{h_{n,2}\right\}_{n},\left\{f_{n,2}^{(i)}\right\}_{n},\left\{g_{n,2}^{(j)}\right\}_{n}\subset H^{\odot 2}$ for $1\leq i\leq k_{1}$ and $1\leq j\leq k_{2}$ for $k_{1}>0$ and $k_{2}>0$. Suppose that the sequences $\left\{I_{2}(f_{n,2}^{(i)})\right\}_{n}$, $\left\{I_{2}(g_{n,2}^{(j)})\right\}_{n}$ and $\left\{I_{2}(h_{n,2})\right\}_{n}$ converge in law to a standard normal variable and that $\langle k_{n,2},k^{\prime}_{n,2}\rangle_{H\otimes H}\to 0$, for any distinct sequences $\left\{k_{2,n}\right\}_{n}$ and $\left\{k^{\prime}_{2,n}\right\}_{n}$ chosen among $\left\{h_{n,2}\right\}_{n},\left\{f_{n,2}^{(i)}\right\}_{n},\left\{g_{n,2}^{(j)}\right\}_{n}$ for $1\leq i\leq k_{1}$ and $1\leq j\leq k_{2}$.

We have then, see [23, Theorem 1], as $n\to\infty$:

[TABLE]

where $Z^{\top}:=(N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}})^{\top}$ has a $k_{1}+k_{2}+1$-dimensional standard normal distribution. The continuous mapping theorem yields that, as $n\to\infty$:

[TABLE]

where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. It is easy to see that the expression in Eq. (3.1) has a representation of the form $I_{4}(\varphi_{n,4})+I_{2}(\varphi_{n,2})+\varphi_{n,0}$ for $\varphi_{n,l}\in H^{\odot l}$, $l=0,2,4$, and $\lim_{n}\varphi_{n,0}=0$. We find with Slutsky’s theorem that $I_{4}(\varphi_{n,4})+I_{2}(\varphi_{n,2})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}aN+\sum_{i=1}^{k_{1}}b_{i}(R_{i}^{2}-1)+\sum_{j=1}^{k_{2}}\left[c_{j}(P_{j}^{2}-1)+d_{j}P_{j}\right]$, as $n\to\infty$.

Remark 3.2*.*

In the present paper we give necessary and sufficient conditions for weak convergence towards target variables as in Eq. (3.2). We illustrate now why this class of target variables is important.

Consider a random variable $X$ living in the sum of the first two Wiener chaoses:

[TABLE]

with $f_{1}\in H$ and $f_{2}\in H^{\odot 2}$. It is known, see [9, Theorem 6.1] or [15, Proposition 2.7.13], that $I_{2}(f_{2})=\sum_{k=1}^{\infty}\alpha_{k}(I_{1}(h_{k})^{2}-1)$ for an orthonormal system $\left\{h_{i}|i\in\mathbb{N}\right\}\subset H$. Suppose from now on that $\alpha_{k}\neq 0$ for $1\leq k\leq N$ and $\alpha_{k}=0$ for every $k\geq N$. Consider the projection of $f_{1}$ on $\textrm{span}(h_{1},\ldots,h_{N})$, then we find $f_{1}=\beta_{1}h_{1}+\ldots+\beta_{N}h_{N}+\beta_{0}h_{0}$, where $\|h_{0}\|_{H}=1$ and $h_{0}\perp\textrm{span }(h_{1},\ldots,h_{N})$. Hence:

[TABLE]

Since $\langle h_{i},h_{j}\rangle_{H}=\delta_{i,j}$, we have that $\left\{I_{1}(h_{0}),\ldots,I_{1}(h_{N})\right\}$ is a set of independent standard normal variables and some of the coefficients $\beta_{0},\ldots,\beta_{N}$ may be equal to 0. The representation of $X$ found in Eq. (3.4) is thus equivalent to the representation in Eq. (3.2):

[TABLE]

where $b_{i}\neq 0,\,c_{j}d_{j}\neq 0$ for $1\leq i\leq k_{1}~{},~{}1\leq j\leq k_{2}$ and $N,R_{i},P_{j}$ are independent standard normal variables living in the first Wiener chaos. A similar argument applied together with the multiplication formula for Wiener integrals shows that $X=I_{1}(f_{1})+I_{2}(f_{2})$ for:

[TABLE]

for a set of orthonormal functions $h_{0},h_{1},\ldots,h_{k_{1}},h_{1}^{\prime},\ldots,h_{k_{2}}^{\prime}$.

If $k_{1}=0$, we set $b_{j}=0$ for every $j$ and the empty sum in the representations above is removed. Notice that an empty product equals 1. We proceed similarly if $k_{2}=0$. If on the other hand $k_{1}\neq 0$, we suppose that $b_{j}\neq 0$ for every $j=1,\ldots,k_{1}$. We proceed similarly if $k_{2}\neq 0$.

The following Lemma shows that random variables with a representation as in Eq. (3.5) extend the class of random variables with a representation as in Eq. (1.2) by adding a (possibly correlated) normal variable.

Lemma 3.3**.**

Consider the following families of random variables:

1. (A)

$X\stackrel{{\scriptstyle\textrm{Law}}}{{=}}aN+\sum_{i=1}^{k_{1}}b_{i}(R_{i}^{2}-1)+\sum_{i=1}^{k_{2}}\left[c_{i}\left(P_{i}^{2}-1\right)+d_{i}P_{i}\right]$, where $N,R_{1},\ldots,R_{k_{1}}$, $P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$ and $b_{i}\neq 0$ for $1\leq i\leq k_{1}$ if $k_{1}>0$ and $c_{i}d_{i}\neq 0$ for $1\leq i\leq k_{2}$ if $k_{2}>0$. 2. (B)

$Y\stackrel{{\scriptstyle\textrm{Law}}}{{=}}aU_{0}+\sum_{i=1}^{n}\lambda_{i}(U_{i}^{2}-1)$* where $\lambda_{i}\neq 0$ for $1\leq i\leq n$ if $n>0$ and $(U_{0},U_{1},\ldots,U_{n})^{\top}$ is a centered normal vector such that $U_{1},\ldots,U_{n}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$.*

*Then class (A) coincides with class (B). In other words every random variable in (A) has a representation as in (B) and vice versa. *

Proof.

1. (1)

Consider $X\stackrel{{\scriptstyle\textrm{Law}}}{{=}}aN+\sum_{i=1}^{k_{1}}b_{i}(R_{i}^{2}-1)+\sum_{i=1}^{k_{2}}[c_{i}(P_{i}^{2}-1)+d_{i}P_{i}]$ as in (A), then, if $a^{2}+\sum_{i=1}^{k_{2}}d_{i}^{2}\neq 0$:

[TABLE]

Define:

[TABLE]

and drop the corresponding terms if $k_{1}=0$ or $k_{2}=0$. After renaming the coefficients, we find the following representation with $n:=k_{1}+k_{2}$:

[TABLE]

and $(U_{0},U_{1},\ldots,U_{n})^{\top}$ is clearly centered and normal since every linear combination $\sum_{i=1}^{n}{\alpha_{i}U_{i}}$ is normal. This last property follows directly from $N,R_{1},\ldots,R_{k_{1}}$, $P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. The definition of $U_{1},\ldots,U_{n}$ yields that $U_{1},\ldots,U_{n}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. If $a^{2}+\sum_{i=1}^{k_{2}}d_{i}^{2}=0$, then $a=d_{1}=\ldots=d_{k_{2}}=k_{2}=0$. The equivalence of both representations is trivial in this case. 2. (2)

Consider $Y\stackrel{{\scriptstyle\textrm{Law}}}{{=}}aU_{0}+\sum_{i=1}^{n}\lambda_{i}(U_{i}^{2}-1)$ as in (B). Since the case $a=0$ is clear, we suppose that $a\neq 0$ and $U_{0}\sim\mathcal{N}(0,1)$. Let $U:=(U_{0},U_{1},\ldots,U_{n})^{\top}\sim\mathcal{N}(0_{(n+1)\times 1},\Sigma)$ for a positive semi-definite matrix $\Sigma$. Since $U_{1},\ldots,U_{n}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$, we have:

[TABLE]

We suppose first that $\det\Sigma=1-\sum_{i=1}^{n}\sigma_{i}^{2}>0$ then $\Sigma=B\,B^{\top}$ for $B$ defined by:

[TABLE]

Consider a $n+1$-dimensional standard normal vector $V^{\top}=(V_{0},V_{1},\ldots,V_{n})^{\top}$, then $U\stackrel{{\scriptstyle\textrm{Law}}}{{=}}BV$. Hence:

[TABLE]

Noticing that some of the covariances $\sigma_{i}$ may be zero, Eq. (3.7) yields the representation (A) after renaming the independent standard normal random variables and the coefficients.

Consider now the case $\det\Sigma=0$, a standard normal vector $(V_{1},\ldots,V_{n})^{\top}$ and define $B\in\mathbb{R}^{(n+1)\times n}$:

[TABLE]

Then $B\,B^{\top}=\Sigma$ and $U\stackrel{{\scriptstyle\textrm{Law}}}{{=}}BV$, hence:

[TABLE]

The statement follows now as above.

∎

Remark 3.4*.*

The random variable $X$ in (A) may lead to a degenerate normal vector $(U_{0},U_{1},\ldots,U_{n})^{\top}$ in (B). In particular $a=k_{1}=0$, $k_{2}=1$ leads to $X\stackrel{{\scriptstyle\textrm{Law}}}{{=}}d_{1}P_{1}+c_{1}(P_{1}^{2}-1)$ and the corresponding normal vector $(U_{0},U_{1})^{\top}$ is degenerate. It can be easily deduced from the proof of Lemma 3.3 that $(U_{0},U_{1},\ldots,U_{n})^{\top}$ is non-degenerate if $a\neq 0$ or $k_{2}=0$. In order to simplify our calculations, we shall consider in this paper target variables of class (A).

We shall need the characteristic function and the cumulants of $X$, defined in Eq. (3.3) and (3.5).

Lemma 3.5**.**

Consider $k_{1},k_{2}\geq 0$ and

[TABLE]

as defined in Eq. (3.3) and (3.5) where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. We have with $\Delta_{l}:=4c_{l}^{2}+d_{l}^{2}$ for the characteristic function $\varphi_{X}$ of $X$:

[TABLE]

where, as usual, an empty product equals 1 and an empty sum equals 0. Moreover $\varphi_{X}$ is the unique solution of the initial value problem $y(0)=1$ and:

[TABLE]

Proof.

Using the characteristic function of the non-central $\chi^{2}$ distribution and the representation:

[TABLE]

we find for the characteristic functions:

[TABLE]

Hence, with the independence of the standard normal random variables:

[TABLE]

Notice that $\frac{d}{dx}\frac{1}{\sqrt{1-2ix\alpha}}=\frac{1}{\sqrt{1-2ix\alpha}}\,\frac{i\alpha}{1-2ix\alpha},$ for every real constant $\alpha$, thus $\varphi^{\prime}_{X}(x)$ equals:

[TABLE]

Multiplying by $\prod_{j=1}^{k_{1}}(1-2ixb_{j})\,\prod_{j=1}^{k_{2}}(1-2ixc_{j})^{2}\neq 0$ yields:

[TABLE]

Thus $\varphi_{X}$ is a solution of the initial value problem. The uniqueness of the solution follows with the Cauchy-Lipschitz theorem since

[TABLE]

is continuous and bounded on every (real) interval.∎

Remark 3.6*.*

Notice that it may be possible to simplify the differential equation if not all coefficients are pairwise different. In Theorem 1.1, 3.8 and 3.11, this simplification may yield a polynomial of smaller degree. For a special case, this problem is discussed in Theorem 4.9 and Remark 4.10. For the rest of this section we shall allow that not all coefficients are pairwise different. In the case of pairwise different coefficients, the differential equation cannot be simplified and the same observation holds for the polynomials in the previously cited theorems.

Lemma 3.7**.**

Consider $k_{1},k_{2}\geq 0$ and

[TABLE]

as defined in Eq. (3.3) and (3.5) where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Then, for $r\geq 2$:

[TABLE]

Proof.

We have $\kappa_{2}(N)=a^{2}$, $\kappa_{r}(N)=0$ for $r>2$ and $\kappa_{r}(R_{j}^{2}-1)=2^{r-1}(r-1)!$. We notice:

[TABLE]

hence for $r\geq 2$:

[TABLE]

Using the formula for the cumulants of the non-central $\chi^{2}$ distribution, we have for $S\sim\mathcal{N}(\mu,1)$:

[TABLE]

thus:

[TABLE]

The result follows now with the independence of the random variables. ∎

We can now prove the first part of our main result: a sufficient criterion for the convergence in law to $X$.

Theorem 3.8**.**

Consider $k_{1},k_{2}\geq 0$ and

[TABLE]

as defined in Eq. (3.3) and (3.5) where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Suppose that at least one of the parameters $a,k_{1},k_{2}$ is non-zero. Consider a sequence $\left\{F_{n}\right\}_{n}$ of non-zero random variables with $F_{n}=\sum_{i=1}^{p}I_{i}(f_{n,i})$ for $p\geq 2$ fixed and $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq p$. Define:

[TABLE]

If the following conditions hold, as $n\to\infty$:

1. (1)

$\displaystyle{\kappa_{r}(F_{n})\to\kappa_{r}(X),\quad\textrm{ for }r=1,\ldots,\textrm{deg}(P)}$, 2. (2)

\displaystyle{\operatorname{\mathbb{E}}\left[\left|\operatorname{\mathbb{E}}\left[\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]\right)\Big{|}F_{n}\right]\right|\right]\to 0},

then $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$.

Proof.

Notice that an empty product equals 1 and an empty sum equals 0. We prove the result for $a\neq 0$, the other case can be treated similarly. We shall use and extend an idea of Nourdin and Peccati, see [13, Paragraph 3.5]. Since $\sup_{n}\kappa_{2}(F_{n})<\infty$, we have with Chebychev’s inequality that the sequence $\left\{F_{n}\right\}_{n}$ is tight. We shall use the following corollary of Prokhorov’s Theorem: Consider a tight sequence $\left\{F_{n}\right\}_{n}$ of random variables. If every subsequence $\left\{F_{n_{k}}\right\}_{k}$ which converges in law has the same limit $Y$, then the initial sequence $\left\{F_{n}\right\}_{n}$ converges in law to $Y$. We consider thus a subsequence $\left\{F_{n_{k}}\right\}_{k}$ which converges in law to some random variable $Y$. We notice that $\lim_{k}\kappa_{2}(F_{n_{k}})=\lim_{n}\kappa_{2}(F_{n})=\kappa_{2}(X)$, hence $\sup_{k}\operatorname{\mathbb{E}}[F_{n_{k}}^{2}]<\infty$. Since $\left\{F_{n}\right\}_{n}$ lives in a fixed finite sum of Wiener chaoses, the hypercontractivity property implies $\sup_{k}\operatorname{\mathbb{E}}[|F_{n_{k}}|^{r}]<\infty$ for every $r\geq 2$. With $F_{n_{k}}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}Y$, as $k\to\infty$, we have $\operatorname{\mathbb{E}}[Y^{2}]=\lim_{k}\operatorname{\mathbb{E}}[F_{n_{k}}^{2}]=\lim_{n}\operatorname{\mathbb{E}}[F_{n}^{2}]=\kappa_{2}(X)\neq 0$. Hence $Y$ is a non-zero random variable and we have $\lim_{k}\kappa_{r}(F_{n_{k}})=\kappa_{r}(Y)$ for every $r$. On the other hand we have $\lim_{n}\kappa_{r}(F_{n})=\kappa_{r}(X)$ for $r=1,\ldots,\textrm{deg}(P)$, thus:

[TABLE]

To simplify the notations and to avoid complicated indices we shall write from now on $\left\{F_{n}\right\}_{n}$ for the subsequence. By the previous corollary, the proof is complete if we can prove that $\varphi_{F_{n}}$ converges to $\varphi_{X}$. We shall prove this by showing that $\varphi_{Y}=\lim_{n}\varphi_{F_{n}}$ solves the initial value problem of Lemma 3.5. This implies that $\varphi_{Y}=\varphi_{X}$ and hence $Y\stackrel{{\scriptstyle\textrm{Law}}}{{=}}X$. The proof is divided in 5 steps:

• •

Step 1: We show that $\varphi^{\prime}_{Y}(0)=0$ and find an alternative representation for $\operatorname{\mathbb{E}}[\exp(ixF_{n})\Gamma_{r}(F_{n})]$.

• •

Step 2: We calculate:

[TABLE]

and:

[TABLE]

• •

Step 3: We calculate $\varphi_{Y}^{\prime}(x)\prod_{j=1}^{k_{1}}(1-2ixb_{j})\prod_{j=1}^{k_{2}}(1-2ixc_{j})^{2}$.

• •

Step 4: We find an expression for:

[TABLE]

• •

Step 5: The proof is completed by showing that the expressions found in the last two steps are equal. Then $\varphi_{Y}$ is the unique solution to the initial value problem in Lemma 3.5, $Y$ and $X$ are thus equal in distribution and $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$.

For the ease of notation, we define $k:=2k_{2}+k_{1}~{},~{}G_{1}(x):=\prod_{j=1}^{k_{1}}(1-2ixb_{j})$ and $G_{2}(x):=\prod_{j=1}^{k_{2}}(1-2ixc_{j})^{2}.$

Step 1. The random variable $Y$ is non-zero and has moments of every order. We notice that for $x=0$, the differential equation of Lemma 3.5 holds for $\varphi_{Y}$ if we can prove $\varphi_{Y}^{\prime}(0)=0$. We notice that $\operatorname{\mathbb{E}}[|F_{n}|]<\infty$, hence $\varphi_{F_{n}}^{\prime}(0)=i\operatorname{\mathbb{E}}[F_{n}]=0$. On the other hand $\lim_{n}\varphi_{F_{n}}^{\prime}(x)=\varphi_{Y}^{\prime}(x)$ for every $x\in\mathbb{R}$ since $\sup_{n}\operatorname{\mathbb{E}}[F_{n}^{2}]<\infty$ and $iF_{n}\exp(ixF_{n})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}iY\exp(ixY)$ by the continuous mapping theorem. We have thus $\varphi_{Y}^{\prime}(0)=\lim_{n}\varphi_{F_{n}}^{\prime}(0)=0$. We suppose now that $x\neq 0$ and calculate $\operatorname{\mathbb{E}}[\exp(ixF_{n})\Gamma_{r}F_{n}]$ for $r\in\mathbb{N}$. For $r\in\left\{0,1\right\}$:

[TABLE]

and for $r>1$:

[TABLE]

Iteration yields:

[TABLE]

Step 2. We calculate now the sum in (3.9). We have:

[TABLE]

With Eq. (3.11) we have thus:

[TABLE]

We calculate the sum in (3.10) using $\operatorname{\mathbb{E}}[\Gamma_{r}(F_{n})]=\kappa_{r+1}(F_{n})/r!$ and $\operatorname{\mathbb{E}}[\Gamma_{0}(F_{n})]=\operatorname{\mathbb{E}}[F_{n}]=0$:

[TABLE]

Step 3. We have thus for the limit of the expression on the right-hand side of Eq. (3.13), as $n\to\infty$:

[TABLE]

We have with $M_{r-1}:=\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]$, as $n\to\infty$:

[TABLE]

hence:

[TABLE]

using Eq. (3.12) and Eq. (3.14), we have with $\lim_{n}\operatorname{\mathbb{E}}[\Gamma_{r}(F_{n})]=\kappa_{r+1}(X)/r!$ for $r=1,\ldots,\textrm{deg}(P)-1$:

[TABLE]

We have used that $\lim_{n}\varphi_{F_{n}}^{\prime}(x)=\lim_{n}\varphi_{Y}^{\prime}(x)$ pointwise. This can be seen using the continuous mapping theorem and $\sup_{n}\operatorname{\mathbb{E}}[F_{n}^{2}]<\infty$. Multiplying Eq. (3.15) by $i(2ix)^{k+1}$ yields:

[TABLE]

Step 4. We have:

[TABLE]

Step 5. We have that $Y\stackrel{{\scriptstyle\textrm{Law}}}{{=}}X$ if and only if the right-hand side of Eq. (3.17) equals $\varphi_{Y}^{\prime}(x)G_{1}(x)G_{2}(x)$ or, using the previous results and Eq. (3.16) in particular, if the following equality holds:

[TABLE]

If $\varphi_{Y}(x)\neq 0$, we can divide by $i\varphi_{Y}(x)$ and compare the coefficients of $x,x^{2},x^{3},\ldots$ on the left- and right-hand side of Eq. (3.18). For this final part of the proof, see Appendix. Considering the previous remarks, this concludes the proof. ∎

Remark 3.9*.*

1. (1)

Notice that the proof of Eq. (3.18) for the general case is rather lengthy and technical. This is basically due to the differential equation derived in Lemma 3.5 from which follows Eq. (3.18). For special cases, such as the case considered in [4], the differential equation simplifies considerably and therefore a relatively simple recurrence for the moments of the target variable can be proved. In [4], the authors have used this recurrence to prove their main result. In the general case however, this recurrence is hard to handle, therefore we have chosen to use differential equations rather than recurrence relations.

We illustrate now how the proof of the crucial equation can be simplified for the class of target variables considered in [4]. Notice that for $a=c_{i}=d_{i}=0$, $k=k_{1}$ and $X\stackrel{{\scriptstyle Law}}{{=}}\sum_{i=1}^{k}b_{i}(N_{i}^{2}-1)$ with $N_{1},\ldots,N_{k}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$, we have to check the following equation:

[TABLE]

where $P(x):=x\prod_{i=1}^{k}(x-b_{i})$. We use the following relation which can be proved by induction over $l$:

[TABLE]

where $1\leq l\leq k-1$ and:

[TABLE]

and $T_{0}^{(j)}:=1$, see Definition 6.1 for details. We have thus for the left-hand side of Eq. (3.19):

[TABLE]

and for the right-hand side of Eq. (3.19):

[TABLE]

This proves that Eq. (3.19) holds.

In the general case, the calculation of $\sum_{r=l}^{\textrm{deg}(P)-2}\frac{P^{(r+2)}(0)}{(r+2)!2^{r+1}}\frac{\kappa_{r-l+2}(X)}{(r-l+1)!}$ is lengthy. The following relations are needed and can be proved by induction over $l$, for the ease of notation define $k^{\prime}:=2k_{2}+k_{1}+1_{[a\neq 0]}$:

[TABLE]

and:

[TABLE]

where $S_{i}$ and $S_{i}^{(j)}$ are defined similarly to $T_{i}^{(j)}$ with the coefficients $b_{1},\ldots,b_{k_{1}}$ replaced by $c_{1},\ldots,c_{k_{2}}$, see Definition 6.1 for details.

In particular, equations (3.21) and (3.22) imply that the proof of Eq. (3.18) is technical. In Proposition 6.2 we present a proof of the latter equation which does not require Eq. (3.21) and (3.22). The coefficients of the polynomials on both sides of Eq. (3.18) are compared directly. This proof still remains complicated and lengthy. As mentioned above, this is due to the form of the differential equation for the characteristic function of the target variable in the general case. 2. (2)

Theorem 3.8 extends [4, Theorem 3.2,(ii) $\to$ (i)] since it holds for a more general set of target random variables $X$ and we have $L^{1}$-convergence instead of $L^{2}$-convergence. 3. (3)

We notice that for the proof it is essential that, as $n\to\infty$:

[TABLE]

Since $|\exp(ixF_{n})|=1$, the triangle inequality shows that it is sufficient to have:

[TABLE]

as $n\to\infty$. This $L^{1}$-convergence is weaker than $L^{2}$-convergence which results typically from the Cauchy-Schwarz inequality, used to control unbounded factors. 4. (4)

Different random variables $X$ may lead to the same polynomial $P$. Let $N$ be a standard normal variable, $X_{1}\stackrel{{\scriptstyle\textrm{Law}}}{{=}}N^{2}-1+d_{1}N$ and $X_{2}\stackrel{{\scriptstyle\textrm{Law}}}{{=}}N^{2}-1+d_{2}N$ with $d_{1}\neq d_{2}$ non-zero, then Theorem 3.8 yields the polynomial $P(x)=x(x-1)^{2}$. A similar observation holds for [16, Theorem 3.4] where the constant $\mu_{0}$ only appears in the limit of the second cumulant. In our case, the condition (1) of Theorem 3.8 discerns both cases. For special cases it may be useful to consider a polynomial which is different from the ‘standard polynomial’ defined in Theorem 3.8. This observation is founded in the initial value problem established in Lemma 3.5. It can be shown that the differential equation in Lemma 3.5 can be simplified if the $b_{i}$ or the $(c_{j},d_{j})$ are not pairwise different. For instance, if $X_{1}\stackrel{{\scriptstyle\textrm{Law}}}{{=}}(N_{1}^{2}-1)+(N_{2}^{2}-1)$ and $N_{1},N_{2}\stackrel{{\scriptstyle\textrm{i.i.d}}}{{\sim}}\mathcal{N}(0,1)$, Lemma 3.5 yields a differential equation which can be simplified to yield [13, Eq. (1.9)].

We proceed now to the proof of the converse of Theorem 3.8. We shall need the following Lemma which generalizes [4, Eq. (3.6)].

Lemma 3.10**.**

Let $P(x):=x^{1+1_{[a\neq 0]}}\prod_{i=1}^{k_{1}}(x-b_{i})\prod_{i=1}^{k_{2}}(x-c_{i})^{2}$ and $X=I_{1}(f_{1})+I_{2}(f_{2})$, see Eq. (3.3), then:

[TABLE]

Proof.

In this proof all the equalities between random variables hold with probability 1. We consider iterated contractions recursively defined as follows for $f_{2}\in H^{\odot 2}$:

[TABLE]

for $p\geq 2$. We use the representation for $X$ given in Eq. (3.3) and Eq. (3.6).

1. (1)

We prove by induction over $r\geq 1$:

[TABLE]

Notice that all iterated contractions on the right hand side of Eq. (3.23) run over $r+1$ functions.

For $r=1$, we have with the stochastic Fubini theorem and the multiplication formula for multiple Wiener integrals:

[TABLE]

The claim follows since the expectation of every multiple Wiener integral is 0. Suppose now that the claim holds for some $r\geq 1$, then:

[TABLE]

Hence Eq. (3.23) follows for $r+1$. 2. (2)

For $r\geq 2$, we have the following equalities:

[TABLE]

where the last equality holds if exactly one of the functions $g_{1},\ldots,g_{r}$ equals $f_{1}$, all remaining functions $g_{i}$ being equal to $f_{2}$. We notice that $(h_{i}\tilde{\otimes}h_{i})\tilde{\otimes}_{1}(h^{\prime}_{j}\tilde{\otimes}h^{\prime}_{j})=0$ and $(h^{\prime}_{i}\tilde{\otimes}h^{\prime}_{i})\tilde{\otimes}_{1}(h^{\prime}_{j}\tilde{\otimes}h^{\prime}_{j})=1_{[i=j]}h^{\prime}_{i}\tilde{\otimes}h^{\prime}_{i}$ and $(h_{i}\tilde{\otimes}h_{i})\tilde{\otimes}_{1}(h_{j}\tilde{\otimes}h_{j})=1_{[i=j]}h_{i}\tilde{\otimes}h_{i}$. Hence $f_{2}\tilde{\otimes}_{1}f_{2}=\sum_{i=1}^{k_{1}}b_{i}^{2}h_{i}\tilde{\otimes}h_{i}+\sum_{i=1}^{k_{2}}c_{i}^{2}h^{\prime}_{i}\tilde{\otimes}h^{\prime}_{i}$ and, generally:

[TABLE]

To prove Eq. (3.25), we suppose first that $g_{2}=f_{1}$ and $g_{1}=g_{3}=\ldots=g_{r}=f_{2}$. Then:

[TABLE]

Since $g_{1}=g_{3}=\ldots=g_{r}=f_{2}$, it is now easy to see that:

[TABLE]

If, on the other hand $g_{1}=\ldots=g_{l}=f_{2}$ for $l>1$, we can use Eq. (3.24) to see that

[TABLE]

and we can proceed as above to see that Eq. (3.25) holds. 3. (3)

Considering Eq. (3.23), it is easy to see that

[TABLE]

lives in the sum of the first two Wiener chaoses. We consider the projection of the random variable above on the first respectively on the second Wiener chaos. We have with Eq. (3.23):

[TABLE]

If $a=0$, we have:

[TABLE]

If $a\neq 0$, we have $P^{\prime}(0)=0$, hence Eq. (3.26) holds in both cases and:

[TABLE]

We have for the projection on the second Wiener chaos with Eq. (3.24):

[TABLE] 4. (4)

Since $Z:=\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(X)-\operatorname{\mathbb{E}}[\Gamma_{r-1}(X)]\right)$ equals $J_{1}(Z)+J_{2}(Z)$, the claim of the Lemma follows now directly.

∎

Theorem 3.11**.**

Consider $k_{1},k_{2}\geq 0$ and

[TABLE]

as defined in Eq.(3.3) and Eq. (3.5) where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Suppose that at least one of the parameters $a,k_{1},k_{2}$ is non-zero. Consider a sequence $\left\{F_{n}\right\}_{n}$ of non-zero random variables with $F_{n}=\sum_{i=1}^{p}I_{i}(f_{n,i})$ for $p\geq 2$ fixed and $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq p$. Define:

[TABLE]

If $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$, then the following limits hold, as $n\to\infty$:

1. (1)

$\displaystyle{\kappa_{r}(F_{n})\to\kappa_{r}(X),\quad\textrm{ for }r=1,\ldots,\textrm{deg}(P)}$, 2. (2)

\displaystyle{\operatorname{\mathbb{E}}\left[\left|\operatorname{\mathbb{E}}\left[\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]\right)\Big{|}F_{n}\right]\right|\right]\to 0}.

Proof.

The proof of this theorem is identical to [4, Theorem 3.2, Proof of (i)$\to$ (iii) ]. It is enough to replace [4, Lemma 3.1] by Lemma 3.10. ∎

Proof of Theorem 1.1

The main result of this paper is now a direct consequence of Theorem 3.8 and Theorem 3.11.∎

Remark 3.12*.*

1. (1)

We notice that, for any sequences of integrable random variables $\left\{F_{n}\right\}_{n}$ and $\left\{G_{n}\right\}_{n}$, we have that $\operatorname{\mathbb{E}}[|F_{n}|]\to 0$ implies $\operatorname{\mathbb{E}}[|\operatorname{\mathbb{E}}[F_{n}|G_{n}]|]\to 0$, as $n\to\infty$:

[TABLE]

Hence, with the notations of Theorem 1.1, a set of sufficient conditions for $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$, is:

1. (1’)

$\displaystyle{\kappa_{r}(F_{n})\to\kappa_{r}(X),\quad\textrm{for }r=1,\ldots,\textrm{deg}(P)}$, 2. (2’)

$\displaystyle{\operatorname{\mathbb{E}}\left[\left|\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]\right)\right|^{\alpha}\right]\to 0}$ ,

where $\alpha\geq 1$. If $\alpha=2$,we can use Eq. (2.1) to calculate the expectation in (2’). 2. (2)

For $\alpha=2$, conditions (1’) and (2’) can be expressed in terms of conditions for contractions. However the resulting conditions are usually complicated as indicates the following example:

Consider $p\geq 2$ even, $a\neq 0$, $b\neq 0$ and a sequence of functions $\left\{f_{n,p}\right\}\subset H^{\odot p}$ such that, as $n\to\infty$:

[TABLE]

Define sets $A_{l}$ and $B_{l}$ for every even integer $l$ with $2\leq l\leq 3p-4$:

[TABLE]

and:

[TABLE]

Define for every even integer $l$ a condition $C_{l}$ by:

[TABLE]

If conditions $C_{l}$ hold for every even integer $l$ with $2\leq l\leq 3p-4$, we have, as $n\to\infty$:

[TABLE]

The proof of this result is omitted, another example for such conditions can be found in [4, Theorem 4.1].

4 Applications

We notice that our results about convergence in law of sequences living in a finite sum of Wiener chaoses can be extended to match convergence in total variation, see for instance [15, Appendix C]. Indeed a direct application of our results together with [4, Lemma 3.3] or [17, Theorem 3.1] proves that we can replace the convergence in law of sequences living in a finite sum of Wiener chaoses by convergence in total variation.

4.1 Recovering classical criteria

As a direct consequence of Theorem 1.1, we get the following Corollary 4.1 which extends [4, Theorem 3.2]. We have proved that (2) is necessary and sufficient for convergence in law whereas the authors in the cited reference need $L^{2}$-convergence of the conditional expectation to prove $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$. As pointed out in Remark 3.9 this results from the Cauchy-Schwarz inequality. Moreover, Corollary 4.1 extends the first part of [6, Proposition 1.7] to more general linear combinations of independent central $\chi^{2}$ distributed random variables. As anticipated, if $b_{1}=\ldots=b_{k}=1$, the polynomial $P$ can be simplified.

Corollary 4.1**.**

Consider $X=I_{2}(f_{2})=\sum_{i=1}^{k}b_{i}(R_{i}^{2}-1)$ for $R_{1},\ldots,R_{k_{1}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$ and $b_{i}\neq 0$ for $i=1,\ldots,k$, see Eq. (3.3) and (3.5). Define $P(x)=x\prod_{j=1}^{k}(x-b_{i})$. Let $\left\{F_{n}\right\}_{n}$ be a sequence of non-zero random variables such that each $F_{n}$ lives in a finite sum of Wiener chaoses, i.e. $F_{n}=\sum_{i=1}^{m}I_{i}(f_{n,i})$ for $n\geq 1$ and $m\geq 2$ fixed, $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq m$. The following two asymptotic relations (1) and (2) are equivalent, as $n\to\infty$:

1. (1)

$F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$; 2. (2)

1. (a)

$\kappa_{r}(F_{n})\to\kappa_{r}(X)$, for $r=2,\ldots,k+1$, 2. (b)

\displaystyle{\operatorname{\mathbb{E}}\left[\left|\operatorname{\mathbb{E}}\left[\sum_{r=1}^{k+1}\frac{P^{(r)}(0)}{r!2^{r-1}}\left(\Gamma_{r-1}(F_{n})-\operatorname{\mathbb{E}}[\Gamma_{r-1}(F_{n})]\right)\Big{|}F_{n}\right]\right|\right]\to 0}.

Proof.

Use Theorem 1.1 with $a=k_{2}=0$. ∎

We come now to the seminal paper [21] of Nualart and Peccati in which the authors have characterised the convergence in law to a standard normal random variable. Since this first paper, the conditions given in [21] have been extended, see for instance [13]. Taking $k_{1}=k_{2}=0$ and $a=1$, we have with Theorem 1.1 the following characterisation which corresponds to condition $(iii^{\prime})$ cited in Section 1.2:

Corollary 4.2**.**

Consider $X=aI_{1}(h_{0})$ for $I_{1}(h_{0})\sim\mathcal{N}(0,1)$ and $a\neq 0$. Let $\left\{F_{n}\right\}_{n}$ be a sequence of non-zero random variables such that each $F_{n}$ lives in a finite sum of Wiener chaoses, i.e. $F_{n}=\sum_{i=1}^{m}I_{i}(f_{n,i})$ for $n\geq 1$ and $m\geq 2$ fixed, $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq m$. The following two asymptotic relations (1) and (2) are equivalent, as $n\to\infty$:

1. (1)

$F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, 2. (2)

1. (a)

$\kappa_{2}(F_{n})\to a^{2}$, 2. (b)

\displaystyle{\operatorname{\mathbb{E}}\left[\left|\operatorname{\mathbb{E}}\left[\langle DF_{n},-DL^{-1}F_{n}\rangle_{H}-a^{2}\Big{|}F_{n}\right]\right|\right]\to 0}.

Proof.

The equivalence of (1) and (2) follows directly from Theorem 1.1 with $P(x)=x^{2}$, since:

[TABLE]

If (2) holds, we have clearly $\lim_{n}\kappa_{2}(F_{n})=a^{2}$. If (1) holds, the latter limit follows from [4, Lemma 3.3]. ∎

Remark 4.3*.*

The equivalence of (1) and (2) in the last corollary can be extended to sequences $\left\{F_{n}\right\}_{n}$ living in $\mathbb{D}^{1,2}$ if some assumptions on the boundedness of the $\Gamma_{1}$-operator is added. Indeed we can prove that (1) implies (2) if $\sup_{n}\operatorname{\mathbb{E}}[\Gamma_{1}(F_{n})^{2}]<\infty$. It is unsure whether the equivalence of (1) and (2) holds without any additional assumptions if $\left\{F_{n}\right\}_{n}\subset\mathbb{D}^{1,2}$.

We use now Remark 3.12 to recover more criteria for the convergence in law, see Corollary 4.4. Notice that (1) is [15, Theorem 5.3.1] for the case of a sequence $\left\{F_{n}\right\}_{n}$ living in a fixed sum of Wiener chaoses, whereas (2) is the sufficient part of the criterion given in [13, Eq. (1.3)] with $L^{2}$-convergence replaced by $L^{1}$-convergence. Since all $L^{p}$-norms inside a fixed sum of Wiener chaoses are equivalent, the criterion given in (2) below is thus necessary and sufficient.

Corollary 4.4**.**

Consider $X=aI_{1}(h_{0})$ for $I_{1}(h_{0})\sim\mathcal{N}(0,1)$ and $a\neq 0$. Let $\left\{F_{n}\right\}_{n}$ be a sequence of non-zero random variables with $\kappa_{2}(F_{n})\to a^{2}$, as $n\to\infty$.

1. (1)

If $F_{n}=\sum_{i=1}^{m}I_{i}(f_{n,i})$ for $n\geq 1$, $m\geq 2$ fixed with $\left\{f_{n,i}\right\}_{n}\subset H^{\odot i}$ for $1\leq i\leq m$ and, as $n\to\infty$:

[TABLE]

then $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$. 2. (2)

If $F_{n}=I_{p}(f_{n,p})$ with $\left\{f_{n,p}\right\}_{n}\subset H^{\odot p}$ and, as $n\to\infty$:

[TABLE]

then $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$.

4.2 New applications

For the rest of the section we suppose that $p$ is even. Notice that if $p$ is odd, we cannot have:

[TABLE]

as $n\to\infty$ for $k_{1}\geq 1$, $\sup_{n}\|f_{n,p}\|_{H^{\otimes p}}<\infty$ and $\sum_{i=1}^{k}b_{i}^{3}\neq 0$. This can be checked using the third cumulant and Eq. (2.1), see [13, Remark 1.3]. More generally, if $p\geq 3$ is odd, we can exclude a large set of possible target random variables by using the fact that all odd-order cumulants are zero, see Remark 4.10 for details.

The following Lemma shall be needed to prove a sufficient criterion based on the convergence of some contractions and cumulants.

Lemma 4.5**.**

Consider $p_{1}$, $p_{2}\geq 1$, $f\in H^{\odot p_{1}}$ and $g\in H^{\odot p_{2}}$. We have for $0\leq r\leq p_{1}\wedge p_{2}$:

[TABLE]

Proof.

The first inequality is a standard result. The first equality follows from [18, Eq. (2.5)]. The next inequality follows from the Cauchy-Schwarz inequality. The last inequalities follow from the following general result:

[TABLE]

which is, again, a consequence of the Cauchy-Schwarz inequality. ∎

Theorem 4.6**.**

Consider $k_{1},k_{2}\geq 0$ and

[TABLE]

as defined in Eq. (3.3) and Eq. (3.5) where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}{\sim}\mathcal{N}(0,1)$. Suppose that at least one of the parameters $a,k_{1},k_{2}$ is non-zero. Consider a sequence $\left\{F_{n}\right\}_{n}$ of non-zero random variables with $F_{n}=I_{p}(f_{n,p})$ for $p\geq 4$ fixed, even, and $\left\{f_{n,p}\right\}_{n}\subset H^{\odot p}$. Define:

[TABLE]

[TABLE]

As $n\to\infty$, we have $I_{p}(f_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$ if the following hold:

[TABLE]

Proof.

1. (1)

We notice that, for $p\geq 4$ even, condition (4.2) is equivalent to:

[TABLE]

see [13, Proposition 3.1]. If $p=2$, we have necessarily $k_{2}=0$ and the convergence in law of sequences $\left\{I_{2}(f_{n,2})\right\}_{n}$ is completely characterised by necessary and sufficient criteria in [16], see Remark 4.7. 2. (2)

We first prove that, except for the contractions $f_{n,p}\tilde{\otimes}_{p/2}^{(r)}f_{n,p}$ for $r=2,\ldots,\textrm{deg}(P)$, all the contractions, appearing in the representation of $M_{r-1}=\Gamma_{r-1}(I_{p}(f_{n,p}))-\operatorname{\mathbb{E}}[\Gamma_{r-1}(I_{p}(f_{n,p}))]$ for $r=2,\ldots,\textrm{deg}(P)$, converge to zero (in the corresponding Hilbert-space norm). With [4, Proposition 2.1], we have for $i\geq 1$ that $\Gamma_{i}(I_{p}(f_{n,p}))-\operatorname{\mathbb{E}}[\Gamma_{i}(I_{p}(f_{n,p}))]$ equals:

[TABLE]

where $S_{i}$ is the set of elements $(r_{1},\ldots,r_{i})$ such that:

[TABLE]

and $c_{p}(r_{1},\ldots,r_{i})$ is defined recursively, see [4, Proposition 2.1]. For $(r_{1},\ldots,r_{i})\in S_{i}\setminus\left\{(p/2,\ldots,p/2)\right\}$, we prove that, as $n\to\infty$:

[TABLE]

If $r_{1}\neq p/2$, we have $\|f_{n,p}\tilde{\otimes}_{r_{1}}f_{n,p}\|_{H^{\otimes(2p-2r_{1})}}\to 0$, as $n\to\infty$, since $r_{1}\neq p$. If $i\geq 2$ and $r_{1}=p/2$, let $r_{1}=\ldots=r_{j}=p/2$ and $r_{j+1}\neq p/2$ with $1\leq j\leq i-1$. We have $r_{j+1}\neq p$, since otherwise the definition of $S_{i}$ yields the following contradictions:

[TABLE]

and:

[TABLE]

Thus, with Lemma 4.5:

[TABLE]

Since $1\leq r_{j+1}\leq p-1$ and $r_{j+1}\neq p/2$, we have that $1\leq p-r_{j+1}\leq p-1$ and $p-r_{j+1}\neq p/2$. Thus $\|f_{n,p}\otimes_{p-r_{j+1}}f_{n,p}\|_{H^{\otimes(2r_{j+1})}}\to 0$, as $n\to\infty$, since $\sup_{n}\|f_{n,p}\|_{H^{\otimes p}}<\infty$, Eq. (4.4) is proved. 3. (3)

With Eq. (4.4), we have for $i\geq 1$, as $n\to\infty$:

[TABLE]

where we have used the definition of $c_{p}$ and [4, Proposition 2.1]. For $i=0$, we have trivially $\operatorname{\mathbb{E}}[(\Gamma_{0}(I_{p}(f_{n,p}))-\operatorname{\mathbb{E}}[\Gamma_{0}(I_{p}(f_{n,p}))]-I_{p}(f_{n,p}))^{2}]=0$. Thus with $M_{r-1}:=\Gamma_{r-1}(I_{p}(f_{n,p}))-\operatorname{\mathbb{E}}[\Gamma_{r-1}(I_{p}(f_{n,p}))]$, as $n\to\infty$:

[TABLE]

On the other hand, as $n\to\infty$:

[TABLE]

The result follows with Eq. (4.5) and (4.6), the reverse triangle inequality, Theorem 1.1 and Remark 3.12.

∎

Remark 4.7*.*

If $p=2$ and $\left\{f_{n,2}\right\}_{n}\subset H^{\odot 2}$, it is known that $I_{2}(f_{n,2})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X^{\prime}$ as $n\to\infty$ implies that $X^{\prime}$ has a representation as in Eq. (1.2) with $k_{1}\leq\infty$ and $k_{2}=0$. In particular, if $k_{1}<\infty$, $k_{2}=a=0$ and $p=2$, Theorem 4.6 holds and is the sufficient part of [4, Proposition 3.1]. It is shown in [4] that the conditions of Theorem 4.6 are also necessary for the case $p=2$. The case $p=2$, $k_{1}<\infty$, $a\neq 0$ and $k_{2}=0$ is not covered by [4] but it can be shown that the conditions given in Theorem 4.6 are also necessary conditions for this case. This can be proved using [16, Theorem 3.4] and [15, Eq. (2.7.17)].

We give now an example for the use of Theorem 4.6 in the case $k_{2}=0$ where the approximating sequence lives in a fixed Wiener chaos. We notice that it is so far unknown whether a sequence living in a fixed chaos can converge to a target variable as in Theorem 4.6 with $k_{2}\neq 0$, see Remark 4.10.

Example 4.8*.*

Consider $p\geq 2$ and sequences of non-zero functions $\left\{f_{n,p}\right\}_{n}$, $\left\{g_{n,p}\right\}_{n}\subset H^{\odot p}$ and $\left\{h_{n,2p}\right\}_{n}\subset H^{\odot(2p)}$ such that the sequences $\left\{I_{p}(f_{n,p})\right\}_{n}$, $\left\{I_{p}(g_{n,p})\right\}_{n}$ and $\left\{I_{2p}(h_{n,2p})\right\}_{n}$ converge in law to a standard normal variable. We also suppose that $\langle f_{n,p},g_{n,p}\rangle_{H^{\otimes p}}\to 0$, as $n\to\infty$. We have then, see [23, Theorem 1], as $n\to\infty$:

[TABLE]

where $(R_{1},R_{2},N)^{\top}$ has a $3$-dimensional standard normal distribution. Using the continuous mapping theorem, [21, Theorem 1] and proceeding as in [13, Proposition 4.1], it is easy to see that, as $n\to\infty$:

[TABLE]

where $\varphi_{n,2p}:=f_{n,p}\tilde{\otimes}f_{n,p}-g_{n,p}\tilde{\otimes}g_{n,p}+h_{n,2p}$. Alternatively the latter convergence in law can be proved using Theorem 4.6. Define:

[TABLE]

We shall need the following limits, as $n\to\infty$:

1. (a)

$\langle f_{n,p}\tilde{\otimes}f_{n,p},g_{n,p}\tilde{\otimes}g_{n,p}\rangle_{H^{\otimes(2p)}}\to 0$, 2. (b)

$\langle f_{n,p}\tilde{\otimes}f_{n,p},h_{n,2p}\rangle_{H^{\otimes(2p)}}\to 0$ and $\langle g_{n,p}\tilde{\otimes}g_{n,p},h_{n,2p}\rangle_{H^{\otimes(2p)}}\to 0$, 3. (c)

$\|(f_{n,p}\tilde{\otimes}f_{n,p})\tilde{\otimes}_{p}(g_{n,p}\tilde{\otimes}g_{n,p})\|_{H^{\otimes(2p)}}\to 0$, 4. (d)

$\|(f_{n,p}\tilde{\otimes}f_{n,p})\tilde{\otimes}_{p}h_{{n,2p}}\|_{H^{\otimes(p)}}\to 0$ and $\|(g_{n,p}\tilde{\otimes}g_{n,p})\tilde{\otimes}_{p}h_{{n,2p}}\|_{H^{\otimes(p)}}\to 0$, 5. (e)

$\|\varphi_{n,2p}\tilde{\otimes}_{p}^{(r)}\varphi_{n,2p}-k_{2p}^{r-1}f_{n,p}\tilde{\otimes}f_{n,p}-k_{2p}^{r-1}g_{n,p}\tilde{\otimes}g_{n,p}\|_{H^{\otimes(2p)}}\to 0$ for $r=2,3,4$, 6. (f)

$\|\varphi_{n,2p}\tilde{\otimes}_{l}\varphi_{n,2p}\|_{H^{\otimes(4p-2l)}}\to 0$ for $1\leq l\leq 2p-1$ and $l\neq p$.

These statements (a)-(f). can be proved using [21, Theorem 1] and [13, Theorem 1.2] We check the conditions of Theorem 4.6.

1. (1)

We prove that $\lim_{n}\kappa_{4}(I_{2p}(\varphi_{n,2p}))=\kappa_{4}(X)$. The convergence of the cumulants of order 2 and 3 can be seen similarly using (a)-(f) and [13, Eq. (3.4)]. We use $\kappa_{4}(I_{2p}(\varphi_{n,2p}))=\operatorname{\mathbb{E}}[I_{2p}(\varphi_{n,2p})^{4}]-3\operatorname{\mathbb{E}}[I_{2p}(\varphi_{n,2p})^{2}]^{2}$ and [13, Eq. (3.6)]:

[TABLE]

The fifth equality follows from [13, Theorem 1.2.(iv)]. For the last equality we have used that, for independent random variables, the cumulant of the sum equals the sum of the cumulants and $\kappa_{r}(N)=0$ for every $r\geq 3$. We have thus proved Eq. (4.1). Notice that Eq. (4.2) holds because of (f). We have $P(x)=x^{2}(x-1)(x+1)$ and:

[TABLE]

We check now Eq. (4.3):

[TABLE]

as $n\to\infty$. Theorem (4.6) yields now $I_{2p}(\varphi_{n,2p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$.

We consider now the convergence in law to random variable with a centered $\chi^{2}$ law and compare the new criterion of Theorem 4.6 with the main result of [13]. We shall see that in this case, both criteria are equivalent.

Theorem 4.9**.**

Consider $k_{1}>0$, $a=k_{2}=0$ and $X=\sum_{i=1}^{k_{1}}(R_{i}^{2}-1)$ as defined in Eq. (3.3) and Eq. (3.5), where $R_{1},\ldots,R_{k_{1}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Consider a sequence $\left\{F_{n}\right\}_{n}$ of non-zero random variables with $F_{n}=I_{p}(f_{n,p})$ for $p\geq 2$ even fixed and $\left\{f_{n,p}\right\}_{n}\subset H^{\odot p}$. Define $P(x)=x(x-1)^{k_{1}}$ and suppose that $\kappa_{2}(F_{n})\to 2k_{1}$, as $n\to\infty$. The following conditions a., b. and c. are equivalent, as $n\to\infty$:

1. a.

$F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, 2. b.

$\displaystyle{\|f_{n,p}\tilde{\otimes}_{l}f_{n,p}\|_{H^{\otimes(2p-2l)}}\to 0,\quad\textrm{ for every }1\leq l\leq p-1\textrm{ with }l\neq p/2,}$, 2. 2)

$\|f_{n,p}\tilde{\otimes}_{p/2}f_{n,p}-2/c_{p}f_{n,p}\|_{H^{\otimes p}}\to 0$** 3. c.

$\displaystyle{\|f_{n,p}\tilde{\otimes}_{l}f_{n,p}\|_{H^{\otimes(2p-2l)}}\to 0,\quad\textrm{ for every }1\leq l\leq p-1\textrm{ with }l\neq p/2,}$, 2. 2)

$\displaystyle{\left\|\sum_{r=1}^{\textrm{deg}(P)}\frac{P^{(r)}(0)}{r!2^{r-1}}f_{n,p}\tilde{\otimes}_{p/2}^{(r)}f_{n,p}c_{p}^{r}\right\|_{H^{\otimes p}}\to 0}$** 3. 3)

$\kappa_{r}(I_{p}(f_{n,p}))\to\kappa_{r}(X),\quad\textrm{for }r=1,\ldots,\textrm{deg }(P).$**

Proof.

The equivalence of a. and b. is proved in [13]. We prove the equivalence of b. and c. for $k_{1}\geq 2$ since b. and c. are clearly equivalent if $k_{1}=1$.

Suppose that b. holds, than a. holds as well and $\sup_{n}\operatorname{\mathbb{E}}[F_{n}^{2}]<\infty$ together with the hypercontractivity property yields that $\sup_{n}\operatorname{\mathbb{E}}[|F_{n}|^{r}]<\infty$ for every $r\geq 1$, thus $\operatorname{\mathbb{E}}[F_{n}^{r}]\to\operatorname{\mathbb{E}}[X^{r}]$ for every $r\in\mathbb{N}$. Hence c.3) holds. We have with $k=k_{1}$:

[TABLE]

where we have used that $\sum_{r=0}^{k}\begin{pmatrix}k\\ r\end{pmatrix}(-1)^{r}=(-1+1)^{k}=0$. We have for $l\geq 2$, as $n\to\infty$:

[TABLE]

hence c.2) follows with the triangle inequality. Suppose now that c. holds. We have to prove that $\|c_{p}^{2}f_{n,p}\tilde{\otimes}_{p/2}^{(2)}f_{n,p}-2c_{p}f_{n,p}\|^{2}_{H^{\otimes p}}\to 0$, as $n\to\infty$, or equivalently:

[TABLE]

We have used in the last step that:

[TABLE]

for all integers $k,l$ with $k\geq 1$ and $l\geq 2$. This can be checked directly using the integral representation of the contractions.

• •

If $k_{1}=2$, c.2) yields that, as $n\to\infty$:

[TABLE]

hence, since $\sup_{n}\|f_{n,p}\|_{H^{\otimes p}}<\infty$, as $n\to\infty$:

[TABLE]

The linearity of the scalar product yields that relation (4.8) holds.

• •

If $k_{1}\geq 3$, the calculations made in the proof of Theorem 4.6, together with [15, Theorem 8.4.4] and c.3) show that:

[TABLE]

for $i=3,\ldots,k_{1}+1$. Thus:

[TABLE]

This completes the proof. ∎

Remark 4.10*.*

1. (1)

The equivalence of a. and b. above is the main result of [13, Theorem 1.2]. Theorem 4.9 shows that, although the polynomials defined throughout this paper may not always be of minimal degree if some of the coefficients are equal, the criterion of Theorem 4.6 is necessary and sufficient in some situations. It is yet unknown if the conditions of Theorem 4.6 are always necessary and sufficient. 2. (2)

Notice that the problem of characterising possible target variables amongst all random variables with a representation as in Eq. (3.5) is far from being solved if the approximating sequence lives in a fixed Wiener chaos. Indeed further research on this topic is needed in order to make a comprehensive statement, but it seems doubtful that $c_{1}(P_{1}^{2}-1)+d_{1}P_{1}$ with $c_{1},d_{1}\neq 0$ is a possible target for a sequence living in a chaos of fixed order. For every sequence living in a Wiener chaos of odd order, all moments of odd order vanish which reduces the class of possible target variables $X=\sum_{i=1}^{k}[c_{i}(P_{i}^{2}-1)+d_{i}P_{i}]$ with $P_{1},\ldots,P_{k}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$ considerably. The following criterion allows to exclude such target variables by just considering the coefficients $c_{i}$ and $d_{j}$. Suppose without loss of generality that the coefficients satisfy the following conditions:

• •

$c_{1},\ldots,c_{m}\in\mathbb{R}^{*}$,

• •

$c_{m+1}=\ldots=c_{k}=0$,

• •

the coefficients $c_{1},\ldots,c_{m}$ are sorted in increasing order of their absolute values and:

[TABLE]

where $k_{0}:=0<k_{1}\leq k_{2}\leq\ldots\leq k_{l}=m$.

Then $I_{p}(f_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$, for $p\geq 3$ odd, implies that:

1. (i)

$m$ is even, 2. (ii)

$k_{i}$ is even for every $i\in\left\{1,\ldots,l\right\}$ and, more precisely:

[TABLE] 3. (iii)

we have for every $i\in\left\{1,\ldots,l\right\}$:

[TABLE]

An analogue version holds for target variables in class (B) of Lemma 3.3.

5 Stable convergence

In this section we consider sequences of non-zero random variables living in a finite sum of Wiener chaoses. The sequences are supposed to converge in law to a non-zero target variable $X$ and we ask whether the sequence also converges stably. Our first result follows from [12, Theorem 1.3]. As before, we shall write $\varphi_{Y}$ for the characteristic function of a random variable $Y$.

Theorem 5.1**.**

Consider $p\geq 3$ and the random variable $X\stackrel{{\scriptstyle\textrm{Law}}}{{=}}aN+\sum_{i=1}^{k_{1}}b_{i}(R_{i}^{2}-1)+\sum_{i=1}^{k_{2}}[c_{i}(P_{i}^{2}-1)+d_{i}P_{i}]$ where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}\stackrel{{\scriptstyle\textrm{i.i.d.}}}{{\sim}}\mathcal{N}(0,1)$. Assume that $\left\{g_{n,p}\right\}_{n}\subset H^{\odot p}$ and, as $n\to\infty$:

[TABLE]

If $I_{p}(g_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$, then:

[TABLE]

or, equivalently, for every $\mathcal{F}$-measurable random variable $Z$:

[TABLE]

where $\mathcal{F}$ is introduced in Section 2.1, and $X$ is independent of the underlying Brownian motion.

Proof.

Consider $f\in H$ with $\|f\|_{H}=1$. We have to prove that, as $n\to\infty$:

[TABLE]

see Section 2.4, where $I_{1}(f)$ is independent of $X$. Since $I_{p}(g_{n,p})$ is a non-zero random variable, we have $\|g_{n,p}\|_{H^{\otimes p}}\neq 0$ for every $n\in\mathbb{N}$. Define $g^{\prime}_{n,p}:=\frac{g_{n,p}}{\sqrt{p!\|g_{n,p}\|_{H^{\otimes p}}^{2}}}$ for every $n$, then $I_{p}(g^{\prime}_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X/\sqrt{A}$, as $n\to\infty$, where $A:=a^{2}+2\sum_{i=1}^{k_{1}}b_{i}^{2}+\sum_{i=1}^{k_{2}}(2c_{i}^{2}+d_{i}^{2})\neq 0$. We have $\operatorname{\mathbb{E}}[I_{p}(g^{\prime}_{n,p})^{2}]=1$ and with Eq. (5.1), as $n\to\infty$:

[TABLE]

[12, Theorem 1.3] yields with $a_{n}:=1/\sqrt{p!\|g_{n,p}\|^{2}_{H^{\otimes p}}}\to a:={1}/{\sqrt{A}}$, as $n\to\infty$:

[TABLE]

We have with $|e^{ix}-1|\leq|x|$ for every $x\in\mathbb{R}$:

[TABLE]

as $n\to\infty$. Hence:

[TABLE]

and with $t^{\prime}_{2}:=t_{2}/\sqrt{A}$ :

[TABLE]

Since $(t_{1},t^{\prime}_{2})\mapsto\varphi_{I_{1}(f)}(t_{1})\varphi_{X}(t^{\prime}_{2})$ is the characteristic function of $(I_{1}(f),X)^{\top}$ where $I_{1}(f)$ is independent of $X$, we have that Eq. (5.3), or equivalently Eq. (5.2) holds. Since $f\in H$ with $\|f\|_{H}=1$ is arbitrary, the statement follows with the remarks of Section 2.4. ∎

Remark 5.2*.*

The proof of the previous theorem was straightforward since the converging sequence of random variables lives in a fixed Wiener chaos. Under these assumptions, [12, Theorem 1.3] yields the desired stable convergence. If the sequence of random variables is allowed to live in a finite sum of Wiener chaoses or if assumption (5.1) does not hold, the conditions ensuring stable convergence involve $\Gamma$-operators.

For the next results, the target variables are (again) supposed to have the form:

[TABLE]

where $N,R_{1},\ldots,R_{k_{1}},P_{1},\ldots,P_{k_{2}}$ are independent standard normal variables and $b_{i}\neq 0$ for $1\leq i\leq k_{1}$, as well as $c_{j}d_{j}\neq 0$ for $1\leq j\leq k_{2}$. We suppose that at least one of the parameters $k_{1},k_{2}$ is positive. We shall use the convention $0^{0}:=1$.

Theorem 5.3**.**

Consider $p\geq 3$, $k_{1},k_{2}\geq 0$ and $X$ as defined in Eq. (5.4). Suppose that $\left\{g_{n,l}\right\}_{n}\subset H^{\odot l}$ for $1\leq l\leq p$ and:

[TABLE]

Define $P(x)=x^{1+1_{[a\neq 0]}}\prod_{i=1}^{k_{1}}(x-b_{i})\prod_{i=j}^{k_{2}}(x-c_{j})^{2}$ and suppose that, as $n\to\infty$:

[TABLE]

If the following two conditions hold, as $n\to\infty$, for every $f\in H$ with $\|f\|_{H}=1$ and $k:=2k_{2}+k_{1}$:

[TABLE]

then $F_{n}\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}X$, as $n\to\infty$, and $X$ is independent of the underlying Brownian motion.

Proof.

Theorem 3.12, Eq. (5.5) and Eq. (5.6) imply that $F_{n}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}X$, as $n\to\infty$. Define:

[TABLE]

The obvious dependence of $(t_{1},t_{2})$ is dropped in the first equalities. The proof is divided in three steps:

• •

we derive two equations involving derivatives of $\varphi_{n}$,

• •

we prove that the sequence $\left\{(I_{1}(f),F_{n})^{\top}\right\}_{n}$ is tight and that we have $(I_{1}(f),F_{n})^{\top}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}(I_{1}(f),X)^{\top}$, as $n\to\infty$, where $X$ is independent of $I_{1}(f)$,

• •

we conclude that $F_{n}\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}X$, as $n\to\infty$, and $X$ is independent of the underlying Brownian motion.

1. (1)

Consider $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{*}$. Then for $r\geq 2$:

[TABLE]

We have used the integration by parts rule and $-\delta DL^{-1}F=F-\operatorname{\mathbb{E}}[F]$ for every $F\in L^{2}(\Omega)$. Repeating this first calculations once again if $r-2\geq 1$, we find:

[TABLE]

and iteration yields finally for $r\geq 1$:

[TABLE]

With $\frac{\partial\varphi_{n}}{\partial t_{2}}(t_{1},t_{2})=i\operatorname{\mathbb{E}}\left[\exp\left(i(t_{1}I_{1}(f)+t_{2}F_{n})\right)F_{n}\right]$, we find:

[TABLE]

Hence, for all $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{*}$:

[TABLE]

With $\Gamma_{1}(I_{1}(f))-1=0$, we find for $(t_{1},t_{2})\in\mathbb{R}^{*}\times\mathbb{R}$:

[TABLE] 2. (2)

We prove that the sequence $\left\{(I_{1}(f),F_{n})^{\top}\right\}_{n}$ is tight. We have with the Markov inequality, for $K\to\infty$:

[TABLE]

Consider a subsequence $\left\{(I_{1}(f),F_{n_{l}})^{\top}\right\}_{l}$ which converges in law to a random vector $V=(V_{1},V_{2})^{\top}$. If we have that $V$ is the same for every converging subsequence and $V\stackrel{{\scriptstyle\textrm{Law}}}{{=}}(I_{1}(f),X)^{\top}$, where $I_{1}(f)$ and $X$ are independent, then we have that $\left\{(I_{1}(f),F_{n})^{\top}\right\}_{n}$ converges in law to the same limit. We have that $\varphi_{n_{l}}(t_{1},t_{2})=\operatorname{\mathbb{E}}\left[\exp\left(i(t_{1}I_{1}(f)+t_{2}F_{n_{l}})\right)\right]\to\varphi_{(V_{1},V_{2})^{\top}}(t_{1},t_{2})=:\varphi^{*}(t_{1},t_{2})$, as $l\to\infty$. Since we have Eq. (5.6) and Eq. (5.8), we have for $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{*}$ and $l\to\infty$ with Eq. (5.9):

[TABLE]

Notice that, as $l\to\infty$, we have $\frac{\partial\varphi_{n_{l}}}{\partial t_{2}}\to\frac{\partial\varphi^{*}}{\partial t_{2}}$ by the continuous mapping theorem and $\kappa_{\beta+1}(F_{n})/\beta!\to k(\beta+1)/\beta!$, as $n\to\infty$, since Eq. (5.5) holds. We have:

[TABLE]

hence:

[TABLE]

To see this, apply Eq. (5.8) with $t_{2}=0$. We find similarly $\frac{\partial\varphi^{*}}{\partial t_{1}}(0,t_{2})=0$. Multiplying Eq. (5.11) by $(it_{2})^{k+1_{[a\neq 0]}}$, we find the following equation for $(t_{1},t_{2})\in\mathbb{R}^{2}$:

[TABLE]

with $\varphi^{*}(0,0)=1$ and $\frac{\partial\varphi^{*}}{\partial t_{1}}(0,t_{2})=\frac{\partial\varphi^{*}}{\partial t_{2}}(t_{1},0)=0$. From Eq. (5.10), we find with $n$ replaced by $n_{l}$ and $l\to\infty$, for $(t_{1},t_{2})\in\mathbb{R}^{*}\times\mathbb{R}$:

[TABLE]

We have used that condition (5.7) implies that $B_{0}$ converges to 0 in Eq. (5.10). Since $\frac{\partial\varphi^{*}}{\partial t_{1}}(0,t_{2})=0$, the differential equation (5.13) holds for every $(t_{1},t_{2})\in\mathbb{R}^{2}$. We have thus a system of partial differential equations, given by Eq. (5.12) and Eq. (5.13), with the conditions $\varphi^{*}(0,0)=1$ and $\frac{\partial\varphi^{*}}{\partial t_{1}}(0,t_{2})=\frac{\partial\varphi^{*}}{\partial t_{2}}(t_{1},0)=0$. We notice that the calculations in the proof of Theorem 3.8, in particular Step 1, Eq. (3.12) and Eq. (3.13) show that the function $(t_{1},t_{2})\mapsto\varphi_{X}(t_{2})$ satisfies Eq. (5.12) and we have $\varphi_{X}(0)=1$ as well as $\varphi^{\prime}_{X}(0)=0$. For a standard normal random variable $M$, the function $(t_{1},t_{2})\mapsto\varphi_{M}(t_{1})$ satisfies Eq. (5.13) and we have $\varphi_{M}(0)=1$ as well as $\varphi^{\prime}_{M}(0)=0$. A solution of the system is given by the function $(t_{1},t_{2})\mapsto\varphi_{M}(t_{1})\varphi_{X}(t_{2})$. Suppose that $\tilde{\varphi}$ is another solution of the system. Define $\Psi(t_{1},t_{2}):=\tilde{\varphi}(t_{1},t_{2})/(\varphi_{M}(t_{1})\varphi_{X}(t_{2}))$. Notice that $\varphi_{M}(t_{1})\varphi_{X}(t_{2})\neq 0$ for $t_{1},t_{2}\in\mathbb{R}$. For $t_{2}\neq 0$, Eq. (5.12) yields an expression of the form $\frac{\partial\tilde{\varphi}}{\partial t_{2}}(t_{1},t_{2})=\tilde{\varphi}(t_{1},t_{2})\,Q(t_{2})$, where $Q$ is a function which depends only on $t_{2}$. Since Eq. (3.12) and Eq. (3.13) hold, we have with the same function $Q$ the representation $\frac{\partial{\varphi_{X}}}{\partial t_{2}}(t_{2})={\varphi_{X}}(t_{2})\,Q(t_{2})$. Hence:

[TABLE]

We find $\frac{\partial\Psi}{\partial t_{2}}(t_{1},0)=0$ since $\frac{\partial\tilde{\varphi}}{\partial t_{2}}(t_{1},0)=\varphi^{\prime}_{X}(0)=0$. A similar calculation shows that $\frac{\partial\Psi}{\partial t_{1}}(t_{1},t_{2})=0$, hence $\nabla\Psi=0$ on $\mathbb{R}^{2}$. We conclude that $\Psi$ is constant and since $\Psi(0,0)=1$, we have that:

[TABLE] 3. (3)

We have finally that every subsequence which converges in law, has the same limit $(I_{1}(f),X)^{\top}$ with $X$ independent of $I_{1}(f)$. Since the $\operatorname{\mathbb{P}}$-completion of the $\sigma$-field generated by $\left\{I_{1}(f):f\in H,\|f\|_{H}=1\right\}$ is the $\sigma$-field $\mathcal{F}$ of the Brownian motion, an application of [11, Lemma 2.3.] concludes the proof, see Section 2.4.

∎

The following Corollary 5.4 is a special case of the implication $(2)\Rightarrow(3)$ in [12, Theorem 1.3].

Corollary 5.4**.**

*Consider $p\geq 3$, $k_{1},k_{2}\geq 0$ and $X$ as defined in Eq. (5.4). Suppose that $\left\{g_{n,p}\right\}_{n}\subset H^{\odot p}$, define $P(x)=x^{1+1_{[a\neq 0]}}\prod_{i=1}^{k_{1}}(x-b_{i})\prod_{j=1}^{k_{2}}(x-c_{j})^{2}$ and suppose that, as $n\to\infty$: *

[TABLE]

If the following condition holds, as $n\to\infty$, for every $f\in H$ with $\|f\|_{H}=1$:

[TABLE]

then $I_{p}(g_{n,p})\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}X$, as $n\to\infty$, and $X$ is independent of the underlying Brownian motion.

Proof.

We prove that Eq. (5.7) and Eq. (5.8) hold, Theorem 5.3 yields then the statement.

1. (1)

We have, as $n\to\infty$:

[TABLE]

We have used the stochastic Fubini theorem for multiple Wiener integrals, and Eq. (5.7) holds since $\left|\exp\left(i(t_{1}I_{1}(f)+t_{2}I_{p}(g_{n,p}))\right)\right|=1$. 2. (2)

To prove that Eq. (5.8) holds, we notice that $\Gamma_{b}(I_{p}(g_{n,p}))$ has a representation as a finite linear combination of random variables of the following form with $R:=\sum_{i=1}^{b}r_{i}$:

[TABLE]

see [4, Proposition 2.1] for details and the set containing $(r_{1},\ldots,r_{b})$ in particular. We can represent $\langle DI_{1}(f),-DL^{-1}\Gamma_{b}(I_{p}(g_{n,p}))\rangle_{H}$ as finite linear combination of terms having the following form:

[TABLE]

• •

Let $A:=(b+1)p-2(r_{1}+\ldots+r_{b})>1$, then we can represent

[TABLE]

as finite linear combination of integrals of the following form:

[TABLE]

where $l$ is a summation index, $Y_{i,l}$ and $X_{j,l}$ represent collections of variables such that:

• –

$\cup_{i=1}^{b+1}Y_{i,l}$ contains $R$ elements and every element is in exactly two of the (non-empty) sets $Y_{1,l},\ldots,Y_{b+1,l}$,

• –

$\left(\cup_{i=1}^{b+1}Y_{i,l}\right)\cap\left\{x_{1},\ldots,x_{A}\right\}=\emptyset$,

• –

$\dot{\cup}_{i=1}^{b+1}X_{i,l}=\left\{x_{1},\ldots,x_{A}\right\}$ .

We have then with $x_{m}\in\left\{x_{1},\ldots,x_{A}\right\}$ and [18, Lemma 2.3], as $n\to\infty$:

[TABLE]

• •

Let $A:=(b+1)p-2(r_{1}+\ldots+r_{b})=1$, then we can represent

[TABLE]

as linear combination of integrals as in Eq. (5.16) and all but one set $X_{i,l}$ is empty. Assume without loss of generality that $X_{1,l}$ is the non-empty set, containing the integration variable, say $x_{0}$. Hence, with [18, Lemma 2.3], as $n\to\infty$:

[TABLE]

Combining these results, we find with $(y_{1}+\ldots+y_{q})^{2}\leq q\left(y_{1}^{2}+\ldots+y_{q}^{2}\right)$ that:

[TABLE]

converges to 0, as $n\to\infty$, hence:

[TABLE]

since $\langle DI_{1}(f),-DL^{-1}\Gamma_{b}({I_{p}(g_{n,p})})\rangle_{H}$ can be represented as linear combination of integrals as in Eq. (5.15). Eq. (5.8) follows since $\left|\exp(i(t_{1}I_{1}(f)+t_{2}I_{p}(g_{n,p})))\right|=1$.

∎

Remark 5.5*.*

Eq. (5.14) holds if $\|g_{n,p}\tilde{\otimes}_{p-1}g_{n,p}\|_{H\otimes H}\to 0$, as $n\to\infty$. This follows directly from Lemma 4.5.

We have the following converse of Theorem 5.3.

Theorem 5.6**.**

With the notations of Theorem 5.3, consider $p\geq 3$, $k_{1},k_{2}\geq 0$ and $X$ as defined in Eq. (5.4). Suppose that $\left\{g_{n,l}\right\}_{n}\subset H^{\odot l}$ for $1\leq l\leq p$, and:

[TABLE]

If condition (5.6) holds and $F_{n}\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}X$, as $n\to\infty$, where $X$ is independent of the underlying Brownian motion, then for every $f\in H$ with $\|f\|_{H}=1$:

1. (1)

Eq. (5.7) and Eq. (5.8) hold, 2. (2)

$\lim_{n}\left(\sum_{l=2}^{p-1}(l+1)!\langle g_{n,l-1}\tilde{\otimes}_{l-1}g_{n,l+1},f\tilde{\otimes}f\rangle_{H\otimes H}+\operatorname{\mathbb{E}}\left[\langle DI_{1}(f),DF_{n}\rangle_{H}^{2}\right]\right)=0$.

Proof.

1. (1)

Consider an arbitrary element $f\in H$ with $\|f\|_{H}=1$. The stable convergence and the independence property imply that $(I_{1}(f),F_{n})^{\top}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}(I_{1}(f),X)^{\top}$, as $n\to\infty$, where $X$ is independent of $I_{1}(f)$. Hence:

[TABLE]

and Eq. (5.9) together with Eq. (3.11) implies that Eq. (5.8) holds for every $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}^{*}$. With Eq. (5.10), we have that Eq (5.7) holds for every $t_{1},t_{2}\neq 0$. To see that Eq. (5.7) holds for $t_{1}=0,t_{2}\neq 0$, calculate the derivative with respect to $t_{1}$:

[TABLE]

hence:

[TABLE]

By the continuous mapping theorem, we have:

[TABLE]

as $n\to\infty$, and $X$ is independent of $I_{1}(f)$. Since $\sup_{n}\operatorname{\mathbb{E}}\left[\left|\exp(it_{2}F_{n})I_{1}(f)\right|^{2}\right]=\sup_{n}\operatorname{\mathbb{E}}[1\,I_{1}(f)^{2}]=1<\infty$, we have:

[TABLE]

thus, as $n\to\infty$:

[TABLE]

We prove that Eq. (5.7) holds for $t_{2}=0,t_{1}\neq 0$:

[TABLE]

as $n\to\infty$.We have used Eq. (2.4) in the last step, the convergence follows as above and Eq. (5.7) holds for $t_{2}=0,t_{1}\neq 0$. In the case $t_{1}=t_{2}=0$, we find with the independence property and Eq. (2.3) that Eq. (5.7) holds.

We see similarly that:

[TABLE]

On the other hand, Eq. (5.8) yields for $t_{2}=0$ the following condition:

[TABLE]

as $n\to\infty$. Since $P^{k+1+1_{[\alpha_{0}\neq 0]}}(0)\neq 0$, the convergence in Eq. (5.18) follows clearly from Eq. (5.17). We conclude that Eq. (5.7) and (5.8) hold for every $(t_{1},t_{2})\in\mathbb{R}\times\mathbb{R}$. 2. (2)

Consider an arbitrary element $f\in H$ with $\|f\|_{H}^{2}=1$. Since $(I_{1}(f),F_{n})^{\top}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}(I_{1}(f),X)^{\top}$, as $n\to\infty$, where $I_{1}(f)\sim\mathcal{N}(0,1)$ is independent of $X$, we have that $(I_{1}(f)^{2},F_{n}^{2})^{\top}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}(I_{1}(f)^{2},X^{2})$, as $n\to\infty$, by the continuous mapping theorem, hence $I_{1}(f)^{2}F_{n}^{2}\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}I_{1}(f)^{2}X^{2}$, as $n\to\infty$. We have that:

[TABLE]

this follows from the hypercontractivity property and the Cauchy-Schwarz inequality. Hence:

[TABLE]

since $I_{1}(f)$ is independent of $X$. With Eq. (5.20):

[TABLE]

and with the stochastic Fubini theorem for multiple Wiener integrals:

[TABLE]

Hence with Eq. (5.21):

[TABLE]

The last equality can be checked directly using the definition of contractions.

∎

Corollary 5.7**.**

Let the notations and assumptions of Theorem 5.6 prevail and suppose that $F_{n}=\sum_{l=1}^{p}I_{l}(g_{n,l})\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}X$, as $n\to\infty$, where $X$ is independent of the underlying Brownian motion. Consider $f\in H$ arbitrary.

If $F_{n}=I_{p}(g_{n,p})$, then we have $\lim_{n}\|g_{n,p}\tilde{\otimes}_{1}f\|_{H^{\otimes(p-1)}}=0$. 2. 2.

If $\lim_{n}\|g_{n,l+1}\tilde{\otimes}_{l-1}g_{n,l-1}\|_{H\otimes H}=0$, for every $2\leq l\leq p-1$, then we have $\lim_{n}\|g_{n,l}\tilde{\otimes}_{1}f\|_{H^{\otimes(l-1)}}=0$, for every $1\leq l\leq p$. 3. 3.

If $F_{n}=I_{p-1}(g_{n,p-1})+I_{p}(g_{n,p})$, then we have $\lim_{n}\|g_{n,p-1}\tilde{\otimes}_{1}f\|_{H^{\otimes(p-2)}}=\lim_{n}\|g_{n,p}\tilde{\otimes}_{1}f\|_{H^{\otimes(p-1)}}=0$.

We can recover [13, Proposition 4.2. and Remark 4.3.].

Proposition 5.8**.**

Consider $p\geq 4$ even and a sequence $\left\{g_{n,p}\right\}_{n}\subset H^{\otimes p}$ such that $I_{p}(g_{n,p})\stackrel{{\scriptstyle\textrm{Law}}}{{\to}}N^{2}-1$, as $n\to\infty$, where $N$ is a standard normal variable. Then $I_{p}(g_{n,p})\stackrel{{\scriptstyle\textrm{st.}}}{{\to}}N^{2}-1$, as $n\to\infty$, and $N^{2}-1$ is independent of the underlying Brownian motion.

Proof.

[13, Theorem 1.2.] implies with $P(x)=x(x-1)$, as $n\to\infty$:

[TABLE]

Moreover the convergence in law implies that $\|g_{n,p}{\otimes}_{p-1}g_{n,p}\|_{H\otimes H}\to 0$, as $n\to\infty$. Notice that in condition (5.8) in Theorem 5.3 we must have $(\alpha,\beta,r)=(1,0,2)$ and condition (5.8) is then satisfied if condition (5.7) holds. We check now condition (5.7).

[TABLE]

as $n\to\infty$. ∎

6 Appendix

Definition 6.1**.**

1. (1)

Consider $k_{1}\geq 0$. If $k_{1}>0$, we suppose that $b_{j}\neq 0$ for every $j=1,\ldots,k_{1}$. We set:

[TABLE]

and for $l=1,\ldots,k_{1}$:

[TABLE]

If $k_{1}=0$, we set $b_{j}=0$ for every $j$, $T_{0}=1$ and $T_{j}=T_{j}^{(l)}=0$ for all other values of $j$ and $l$. 2. (2)

Consider $k_{2}\geq 0$. If $k_{2}>0$, we suppose that $c_{j}d_{j}\neq 0$ for every $j=1,\ldots,k_{2}$. We set:

[TABLE]

and for $l=1,\ldots,k_{2}$:

[TABLE]

If $k_{2}=0$, we set $c_{j}=d_{j}=0$ for every $j$, $S_{0}=1$ and $S_{j}=S_{j}^{(l)}=0$ for all other values of $j$ and $l$.

Proposition 6.2**.**

Proof of Eq. (3.18).

Proof.

We prove that Eq. (3.18) holds for $x\neq 0$. Suppose that $x\varphi_{Y}(x)\neq 0$ and divide Eq. (3.18) by $i\varphi_{Y}(x)$:

[TABLE]

We determine an alternative representation for $P(x)$ and calculate $P^{(l)}(0)/l!$. We have with $k=2k_{2}+k_{1}$

[TABLE]

If for instance $k_{1}=0$, we have $T_{j}=1_{[j=0]}$ and the formula above holds also for the cases $k_{1}=0$, $k_{2}=0$ and $k_{1}=k_{2}=0$. Hence:

[TABLE]

We compare now the terms in $x$, on the left-hand side of Eq. (6.1) we have:

[TABLE]

and for the right-hand side of Eq. (6.1):

[TABLE]

the desired equality for the terms in $x$ follows. The term in $x^{k+1}$ on the left-hand side of Eq. (6.1) is $2^{k}(ix)^{k+1}(-1)^{k_{1}}a^{2}\prod_{j=1}^{k_{2}}c_{j}^{2}\prod_{j=1}^{k_{1}}b_{j}$, and on the right-hand side of Eq. (6.1) we have:

[TABLE]

We have for the term in $x^{m}$ and $1<m<k+1$ on the left-hand side of Eq. (6.1):

[TABLE]

We use the relations $S_{j}=S_{j}^{(l)}+c_{l}S_{j-1}^{(l)}$ and $T_{j}=T_{j}^{(l)}+b_{l}T_{j-1}^{(l)}$ if $k_{1},k_{2}>0$, and set $Z_{j}^{(l)}:=\sum_{j_{2}+j_{3}=j}S_{j_{2}}^{(l)}S_{j_{3}}^{(l)}$. We consider the powers of $b_{j}$ in Eq. (6.2) and suppose that $k_{1}\neq 0$ for the next calculation.

[TABLE]

hence:

[TABLE]

Notice that, with our settings, the previous equality holds also if $k_{1}=0$ or $k_{2}=0$ We consider now the powers of $c_{j}$ in Eq. (6.2) and suppose that $k_{2}>0$ for the next calculation.

[TABLE]

Hence the expression in Eq. (6.3) equals:

[TABLE]

The resulting equality holds also if $k_{1}=0$ or $k_{2}=0$. In the latter case, we have $S_{j_{2}}^{(l)}=c_{j}=d_{j}=0$ and an empty sum in (6.3) equals 0. For the terms of the form $c_{l}^{r}d_{l}^{2}$ in Eq. (6.2), we can make a similar calculation. For the calculation below, we suppose that $k_{2}\neq 0$.

[TABLE]

The resulting equality holds also if $k_{1}=0$ or $k_{2}=0$. We have finally for the term in $x^{m}$ and $1<m<k+1$ on the left-hand side of Eq. (6.2):

[TABLE]

Consider now the term in $x^{m}$ for $1<m<k+1$ on the right-hand side of Eq. (6.1). Define:

[TABLE]

$Q_{m}$ is thus the projection of a series (or a polynomial) on its term of degree $m$. The terms in $x^{m}$ for $1<m<k+1$ on the right-hand side of Eq. (6.1) are given by:

[TABLE]

We calculate now the projections appearing in the expression above.

[TABLE]

Notice that these equalities hold if $k_{1}\geq 0$ and $k_{2}\geq 0$. The following equalities can be derived as above:

[TABLE]

We have thus for expression (6.5):

[TABLE]

This equality holds (again) for $k_{1}\geq 0$ and $k_{2}\geq 0$. Comparing Eq. (6.4) and Eq. (6.6), we find the desired equality of the terms in $x^{m}$ on the left-hand and the right-hand side of Eq. (6.1).∎

Acknowledgements

I would like to thank two anonymous referees for their comments and remarks, as well as Prof. Dr. G. Peccati and Prof. Dr. A. Thalmaier for their time and many interesting discussions and suggestions.