Fourth moment theorems on the Poisson space: analytic statements via product formulae

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carr\'e-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by D\"obler and Peccati (2018) and D\"obler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

1. Introduction 1.1. Overview. A "fourth moment theorem" (FMT) is a probabilistic statement, implying that a certain sequence of (centred and normalised) random variables verifies a central limit theorem (CLT) as soon as the sequence of their fourth moments converges to 3, that is, to the fourth moment of the one-dimensional standard Gaussian distribution. Well-known examples of FMTs are de Jong-type theorems for degenerate U-statistics [DP16], as well as the class of CLTs for multiple stochastic integrals with respect to Gaussian fields discussed e.g. in [NP12,Chapter 5]. The aim of the present paper is to provide an analytical study of FMTs for sequences of multiple stochastic integrals with respect to a general Poisson random measure.
Our starting point will be the recent paper [DP18b], where we proved quantitative FMTs for multiple Wiener-Itô integrals with respect to a general Poisson random measure under mild regularity assumptions for the involved random variables. Such regularity assumptions were later removed in [DVZ18], where the above mentioned fourth moment results were also extended to the multivariate case.
The common scheme in the proofs of the main results in both papers [DP18b,DVZ18] was the implementation of the so-called spectral viewpoint, initiated in the seminal paper [Led12] in the framework of invariant measures of a Markov diffusion generator L. In particular, the two references [DP18b,DVZ18] amply demonstrated that the use of carré-du-champ operators continues to be the right tool for deriving effective quantitative fourth moment bounds, even when dealing with the non-diffusive Ornstein-Uhlenbeck generator L defined on configuration spaces (see the discussion below for precise definitions).
Remarkably, the use of carré-du-champ operators also entails that neither the results of [DP18b] nor those of [DVZ18] rely on the classical product formula for multiple integrals on the Poisson space (see e.g. [Las16,Proposition 5]), whereas product formulae on Gaussian spaces are pivotal in the original proof of the FMT for multiple integrals with respect to an isonormal Gaussian process -see e.g. in [NP12,Chapter 5]. In particular, neither the proofs nor the statements contained in [DP18b,DVZ18] exploited the properties of contraction operators, naturally emerging in connection with multiple stochastic integrals whenever one deals with the computation of higher moments.
It is important to notice that, in the Gaussian situation (see e.g. [NP12, Theorem 5.2.7] and the discussion therein) and differently from what is currently known on the Poisson space, the conditions for the asymptotic normality of a sequence of multiple integrals belonging to a fixed chaos can be phrased equivalently in terms of fourth moments, of norms of contraction kernels, and of variances of squared norms of Malliavin derivatives. As one can deduce by inspecting the hundreds of references listed on the webpage [Web], conditions expressed in terms of contraction kernels are very often (and by far) the easiest to check and the most amenable to analysis, whereas the exact computation of fourth moments conditions often involves a number of combinatorial difficulties.
The main objective of the present paper is to complement the fourth moment conditions from [DP18b,DVZ18] with conditions expressed in terms of norms of contractions and in terms of variances of carré du champ operators, with the precise aim of making the results of [DP18b,DVZ18] even more amenable to analysis. Under mild uniform integrability requirements, we will show that all these conditions are equivalent to the asymptotic normality of the sequence of multiple integrals. Our main findings appear below in Theorem 3.1 (one-dimensional case) and Theorem 3.4 (multi-dimensional case).
In order to establish our results, we will need to review the product formula for multiple Wiener-Itô integrals on the Poisson space. In its standard form (see e.g. [Las16, Proposition 5]), such a formula needs additional L 2 -integrability assumptions on some family of contraction kernels. In order to be able to work in full generality, we will prove below a new general product formula that, under the minimal assumptions of the FMT of [DVZ18], ensures that certain linear combinations of contraction kernels are automatically in the corresponding L 2 spaces. On the other hand, by proving also a converse statement, we obtain a new test criterion, ensuring that the product of two multiple integrals is in L 2 . Such a criterion is only expressed in terms of linear combinations of contractions kernels. All our new findings on this matter are contained in Theorem 2.2.
For the rest of the paper, every random object is defined on a common probability space (Ω, F , P), with E denoting expectation with respect to P. We use the symbol D −→ in order to indicate convergence in distribution of random variables.

A new general product formula on the Poisson space
We start by providing a rough description of the technical setup, that is needed in order to state and understand our main results. We refer to Section 4, as well as to [Las16], for precise definitions and more detailed discussions.
Let us fix an arbitrary measurable space (Z, Z ) endowed with a σ-finite measure µ. Furthermore, we let Z µ := {B ∈ Z : µ(B) < ∞} and denote by η = {η(B) : B ∈ Z } a Poisson random measure on (Z, Z ) with control µ, defined on a suitable probability space (Ω, F , P). We recall that this means that: (i) for each finite sequence B 1 , . . . , B m ∈ Z of pairwise disjoint sets, the random variables η(B 1 ), . . . , η(B m ) are independent, and (ii) that for every B ∈ Z , the random variable η(B) has the Poisson distribution with mean µ(B). Here, we have extended the family of Poisson distributions to the parameter region [0, +∞] in the usual way. For B ∈ Z µ , we also writeη(B) := η(B) − µ(B) and denote bŷ the compensated Poisson measure associated with η. Throughout this paper we will assume that F = σ(η). In order to state our main results, we have to briefly recall the following objects, arising in the context of stochastic analysis for Poisson measures. By L, we denote the generator of the Ornstein-Uhlenbeck semigroup associated with η, and by dom L ⊆ L 2 (P) we denote its domain. It is well-known that −L is a symmetric, diagonalizable operator on L 2 (P) which has has the pure point spectrum N 0 = {0, 1, . . . }. Closely connected to L is the symmetric, bilinear and nonnegative carré du champ operator Γ, which is defined by for all F, G ∈ dom L such that also F G ∈ dom L. For p ∈ N 0 we denote by C p := ker(L + pId) the so-called p-th Wiener chaos associated with η. Here, we denote by Id the identity operator on L 2 (P). It is a well-known fact that, for p ∈ N, the linear space C p coincides with the collection of multiple Wiener-Itô integrals I p (f ) of order p with respect toη. Here, f ∈ L 2 (µ p ) is a square-integrable function on the product space (Z p , Z ⊗p , µ p ). Moreover, for a constant c ∈ R we let I 0 (c) := c. Then, since the kernel of L coincides with the constant random variables, we also have C 0 = {I 0 (c) : c ∈ R}. Multiple integrals have the following two fundamental properties. Let p, q ≥ 0 be integers: then, 1) I p (f ) = I p (f ), wheref denotes the canonical symmetrization of f ∈ L 2 (µ p ), i.e., with S p the symmetric group acting on {1, . . . , p}, we havẽ f (z 1 , . . . , z p ) = 1 p! π∈Sp f (z π(1) , . . . , z π(p) ) ; 2) I p (f ) ∈ L 2 (P), and E I p (f )I q (g) = δ p,q p! f ,g , where δ p,q denotes Kronecker's delta symbol. The Hilbert subspace of L 2 (µ p ) composed of µ p -a.e. symmetric kernes will henceforth be denoted by L 2 s (µ p ). It is a crucial fact that every F ∈ L 2 (P) admits a unique representation where f p ∈ L 2 s (µ p ), p ≥ 1, are suitable symmetric kernel functions, and the series converges in L 2 (P). Identity (2.1) is referred to as the chaotic decomposition of the functional F ∈ L 2 (P). Hence, multiple integrals are in a way the basic building blocks of the space L 2 (P). Note that (2.1) can equivalently be written as where the sum on the right hand side is furthermore orthogonal.
The following analytic notion of a contraction kernel will also be crucial for the statements of our results. Fix integers p, q ≥ 1 as well as symmetric kernels f ∈ L 2 s (µ p ) and g ∈ L 2 If l = 0 and 0 ≤ r ≤ p ∧ q, then we let In particular, if l = r = 0, then f ⋆ 0 0 g = f ⊗ g reduces to the tensor product of f and g, given by (f ⊗ g)(t 1 , . . . , t p , s 1 , . . . , s q ) := f (t 1 , . . . , t p ) · g(s 1 , . . . , s q ) .
More generally, whenever 0 ≤ l = r ≤ p ∧ q, then on customarily writes f ⊗ r g for f ⋆ r r g. In this case, a simple application of the Cauchy-Schwarz inequality shows that f ⊗ r g ∈ L 2 (µ p+q−2r ). This however, does not hold for l < r, in general. For instance, if and only if f ∈ L 4 (µ p ). As shown in our paper [DP18a], the contraction kernel f ⋆ r l g is always µ p+q−r−l -a.e. well-defined as a function on Z p+q−r−l .
Contraction kernels play a major role in this article because they naturally arise in the following classical product formula on the Poisson space that is taken from [Las16]. It was first proved under less general conditions in [Sur84].
Proposition 2.1 (Classical product formula). Let p, q ≥ 1 be integers and assume that f ∈ L 2 s (µ p ) and g ∈ L 2 s (µ q ). If, for all integers 0 ≤ r ≤ p ∧ q and 0 ≤ l ≤ r − 1 one has that f ⋆ l r g ∈ L 2 (µ p+q−r−l ), then Note that the sum appearing on the right hand side of (2.2) is not orthogonal, since it is not arranged according to the orders of the integrals. Introducing the parameter m = r + l, satisfying the restrictions we can rewrite (2.2) as follows: where the symmetric kernel h p+q−m , 0 ≤ m ≤ 2(p ∧ q), is given by From the classical product formula given in Theorem 2.1 we can conclude the validity of (2.3) only under the assumption that all the contraction kernels f ⋆ r l g, 0 ≤ l < r ≤ p ∧ q, be in L 2 which implies that also the h p+q−m are in L 2 . Note also that, under such a restriction, the sum in (2.3) is orthogonal.
The first theoretical result of this paper is a new general product formula, showing that (2.2) continues indeed to hold under the minimal assumption that I p (f )I q (g) ∈ L 2 (P). In particular, this implies that h p+q−m as given in (2.4) is always in L 2 (µ p+q−m ) even though this might not be the case for the individual contractions f ⋆ m−r r g appearing in the defining equation (2.4). Moreover, we also show that the converse is true as well, i.e. that I p (f )I q (g) ∈ L 2 (P) whenever each kernel h p+q−m is in L 2 (µ p+q−m ).
Theorem 2.2 (General product formula on the Poisson space). Suppose that p, q ≥ 1 are integers, select f ∈ L 2 s (µ p ) and g ∈ L 2 s (µ q ), and define F : . In this case, F G has the finite chaotic decomposition (2.5) Remark 2.3. (a) An immediate consequence of Theorem 2.2 is that the product of two multiple integrals with respect toη either has a finite chaos decomposition of order at most p + q or none at all. To the best of our knowledge this fact has not been noted so far. (b) Identity (2.5) in particular holds, whenever F, G ∈ L 4 (P). (c) A similar product formula as in Theorem 2.2, but under less general conditions, can be found in [Pri09, Chapter 6].
3. An extension of the fourth moment theorem In the recent paper [DP18b], under mild integrability conditions, we proved the bounds where F = I p (f ) ∈ C p is a multiple Wiener-Itô integral with respect to η such that E[F 2 ] = 1, N is a standard normal random variable and d W and d K denote the Wasserstein and Kolmogorov distances, respectively (see e.g. [NP12, Appendix C] and the references therein). The above mentioned integrability conditions could be successfully removed for the Wasserstein bound (3.1) in the paper [DVZ18]. In particular, one has the following sequential FMT: Suppose that, for each n ∈ N, p n ≥ 1 is an integer, f n ∈ L 2 s (µ pn ) is a kernel and F n = I pn (f n ) is a multiple integral such that lim In what follows, we will state and prove a substantial refinement of such a result, in the crucial case of a sequence of multiple stochastic integrals belonging to a fixed chaos, that is, such that p n ≡ p ≥ 1. In order to do this, for n ≥ 1, we need to define the auxiliary kernels The next statement is one of the main achievements of the paper.
Theorem 3.1 (Extended fourth moment theorem). Fix an integer p ≥ 1 and let F n = I p (f n ), n ∈ N, be a sequence in C p such that lim n→∞ E[F 2 n ] = 1, with f n ∈ L 2 s (µ p ) for n ∈ N. Let N ∼ N(0, 1) be a standard normal random variable. Consider the following conditions: 2p−m 2 = 0 for all m = 1, . . . , 2p − 1 and lim n→∞ f n ⊗ r f n 2 = 0 for all r = 1, . . . , p − 1.
Remark 3.2. It is very interesting and quite surprising that condition (iii) and the seemingly weaker condition (v) in Theorem 3.1 are indeed equivalent.
(ii)⇒(iv): By Lemma 3.1 of [DP18b] we have the inequality which immediately implies the claim.
(iv)⇒(ii): By Part 2 of [DVZ18, Remark 5.2] (that we can apply, since (iv) ensures that E[F 4 n ] < ∞ for large n) we have the bound proving the implication.
(iv)⇒(v): Using Theorem 2.2, from the computations on page 1895 in [DP18b] (that we can apply since, under (iv), one has that E[F 4 n ] < ∞ for large n), we have and the desired implication follows.
(v)⇒(iv): Using again Theorem 2.2, we see that, under (v), F n ∈ L 4 (P) for n large enough, so that we can apply once again (3.6) to deduce the claim.
(ii)⇒(i): This is an immediate consequence of the fourth moment bound (3.1).
Finally, assume that the sequence (F 4 n ) n∈N is uniformly integrable and that (i) holds.
Furthermore, suppose that V = (V (i, j)) 1≤i,j≤d is a nonnegative definite matrix such that the covariance matrix Σ n of F (n) converges to V as n → ∞. Also, suppose that N = (N 1 , . . . , N Under the assumptions of Proposition 3.3, for n ∈ N and k = 1, . . . , d, define the symmetric kernels h With this notation at hand we can state our multivariate extended fourth moment theorem: s (µ p k ), k = 1, . . . , d. Furthermore, suppose that V = (V (i, j)) 1≤i,j≤d is a nonnegative definite matrix such that the covariance matrix Σ n of F (n) converges to V as n → ∞. Suppose that N = (N 1 , . . . , N d ) T is a d-dimensional centered Gaussian vector with covariance matrix V . Consider the following conditions:   Proof. The equivalence of items (ii)-(v) can be proved similarly as in the proof of Theorem 3.1. One only has to extend the arguments to general positive variances of the coordinates of N. That (ii) implies (i) follows from Theorem 3.3. Finally, the fact that (i) implies (ii) under the assumption of uniform integrability follows as in the previous proof.

General Poisson point processes and technical framework
Here we describe our theoretical framework by adopting the language of [Las16] (see also [LP17]).
Let (Z, Z ) be an arbitrary measurable space endowed with a σ-finite measure µ and denote by N σ = N σ (Z) the space of all σ-finite point measures χ on (Z, Z ) that satisfy χ(B) ∈ N 0 ∪ {+∞} for all B ∈ Z . This space is equipped with the smallest σ-field N σ := N σ (Z) such that, for each B ∈ Z , the mapping N σ ∋ χ → χ(B) ∈ [0, +∞] is measurable. It is convenient to view the Poisson process η as a random element of the measurable space (N σ , N σ ), defined on an abstract probability space (Ω, F , P). Without loss of generality we may assume that F = σ(η). Moreover, we denote by F(N σ ) the class of all measurable functions f : N σ → R and by L 0 (Ω) := L 0 (Ω, F ) the class of real-valued, measurable functions F on Ω. Note that, as F = σ(η), each F ∈ L 0 (Ω) can be written as F = f(η) for some measurable function f. This f, called a representative of F , is P η -a.s. uniquely defined, where P η = P • η −1 is the image measure of P under η on the space (N σ , N σ ). For F = f(η) ∈ L 0 (Ω) and z ∈ Z we define the add one cost operators D + z , z ∈ Z, by . It is straightforward to verify the following product rule: For F, G ∈ L 0 (Ω) and z ∈ Z one has More generally, if m ∈ N and z 1 , . . . , z m ∈ Z, then we define inductively D (1) which shows that the mapping Ω × Z m ∋ (ω, z 1 , . . . , z m ) → D (m) For an integer p ≥ 1 we denote by L 2 (µ p ) the Hilbert space of all square-integrable and real-valued functions on Z p and we write L 2 s (µ p ) for the subspace of those functions in L 2 (µ p ) which are µ p -a.e. symmetric. Moreover, for ease of notation, we denote by · 2 and ·, · the usual norm and scalar product on L 2 (µ p ) for whatever value of p. We further define L 2 (µ 0 ) := R. For f ∈ L 2 (µ p ), we denote by I p (f ) := I η p (f ) the multiple Wiener-Itô integral of f with respect toη. If p = 0, then, by convention, I 0 (c) := c for each c ∈ R. We refer to Section 3 of [Las16] for a precise definition and the following basic properties of these integrals in the general framework of a σ-finite measure space (Z, Z , µ). If F = I p (f ) for some p ≥ 1 and f ∈ L 2 s (µ p ), then for all z ∈ Z one has D + z F = I p−1 f (z, ·) . (4.3) In particular, D + z F is a multiple Wiener-Itô integral of order p − 1. If, on the other hand, p = 0, then it is easy to see that D + z F = 0.
As recalled above, for p ≥ 0 the Hilbert space consisting of all random variables I p (f ), f ∈ L 2 (µ p ), is called the p-th Wiener chaos associated with η, and is customarily denoted by C p .
From Theorem 2 in [Las16] (which is Theorem 1.3 from the article [LP11]) it is known that, for all F ∈ L 2 (P) and all p ≥ 1, the kernel f p in (2.1) is explicitly given by Identity (4.4) will be an essential tool for the proof of Theorem 2.2.

Proof of the product formula
For the proof of Theorem 2.2 we will need the following auxiliary result that provides us with a sufficient condition for an integrable random variable F to be in L 2 (P).
Note that we have used assumption (a) for the last equality and (b) in order to use (5.1) iteratively as well as for the final inequality.
We now provide a detailed proof of Theorem 2.2, which has a purely combinatorial nature and does not make any use of recursive arguments.
Proof of Theorem 2.2. We first make the following important observation: whenever F = I p (f ) and G = I q (g) are as in the statement of the theorem, for all k ∈ N 0 and all fixed z 1 , . . . , z k ∈ Z we have that D (k) z 1 ,...,z k (F G) is a (finite) linear combination of products of two multiple Wiener-Itô integrals of orders less than p and q, respectively. This easily follows iteratively from (4.1) and (4.3). In particular, all summands appearing in this linear combination and, a fortiori, the quantity D (k) z 1 ,...,z k (F G) itself is in L 1 (P). This observation will be used implicitly in the rest of this proof.
Let us now assume first that F G ∈ L 2 (P). Then, we know that a chaotic decomposition of the form In [DP18b, Lemma 2.4] we already proved that h k = 0 for all k ≥ p + q. However, this will also easily follow from the arguments used in the present proof. From (4.4) we immediately get that In order to get more explicit expressions for the h k we introduce the following operators: For a pair (X, Y ) ∈ L 0 (Ω) × L 0 (Ω) and z ∈ Z, define . More generally, if W = (W 1 , . . . , W m ) ∈ {L, R, B} m is a word of length |W | = m in the alphabet {L, R, B} and z 1 , . . . , z m ∈ Z, then we let , if m = 1 and, for m ≥ 2, we define inductively Then, with the multiplication operator K : L 0 (Ω) × L 0 (Ω) → L 0 (Ω) defined by K(X, Y ) := X · Y the product rule (4.1) implies that where the sum runs over all words W of length k. Hence, (5.2) can be written as z 1 ,...,z k (F, G) . ). In what follows we will write W for the equivalence class of the word W . Note that the random quantity K D [W ] z 1 ,...,z k (F, G) is either equal to 0, if l(W ) + b(W ) > p or if r(W ) + b(W ) > q, or else is a product of quantities of the type (p) p−s I s (u) and (q) q−r I t (v), where I s (u) and I t (v) are two multiple integrals, s = p − l(W ) − b(W ), t = q − r(W ) − b(W ) and the kernels u ∈ L 2 (µ s ) and v ∈ L 2 (µ t ) depend on the variables z 1 , . . . , z k (as usual, we use the notation (n) m = n(n − 1) · . . . · (n − m + 1) = n! (n−m)! to indicate the falling factorial, defined for integers 0 ≤ m ≤ n).
Note that, by orthogonality of multiple integrals of different orders, we have According to these facts, let us fix 0 ≤ k ≤ p+q as well as a word W = (W 1 , . . . , W k ) ∈ {L, R, B} k of characteristic χ(W ) = (l, r, b) such that p ≥ l + b, q ≥ r + b and p − l = q − r. Note that we have the identity k = l + r + b. We now aim at expressing in a more explicit way. Firstly, it is clear from the definitions and from (4.2) that h W is a symmetric function of the variables z 1 , . . . , z k ∈ Z. Indeed, we claim that In order to see this, for σ, π ∈ S k , we define the following relation: Let Then, we write σ ≈ π if and only if (σ•π −1 )(B j ) = B j for all j = 1, 2, 3. Equivalently, σ ≈ π if, and only if, for all i ∈ [k] and all j = 1, 2, 3 it holds that σ(i) ∈ B j ⇔ π(i) ∈ B j . This clearly defines an equivalence relation on S k . It is easily checked that for σ ∈ S k its equivalence class [σ] has cardinality |[σ]| = |B 1 |! · |B 2 |! · |B 3 |! = b!l!r! and, hence, there are exactly m := k! b!l!r! equivalence classes. Let σ 1 , . . . , σ m be a complete system of representatives for the relation ≈. It is easy to see that  (1) , . . . , z σ i (k) ) . (5.6)