Vector-valued semicircular limits on the free Poisson chaos

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).


Introduction and background
Let {W t : t ≥ 0} be a standard Brownian motion on R + and let n ≥ 1 be an integer. Denote by I W n (f ) the multiple stochastic Wiener-Itô integral of order n of a symmetric function f ∈ L 2 R n + . Denote by L 2 s R n + the subset of L 2 R n + composed of symmetric functions. The collection of random variables I W n (f ) : f ∈ L 2 s R n + is what is usually called the n-th Wiener chaos associated with W .
In a seminal paper of 2005, Nualart and Peccati [8] proved that convergence to the standard normal distribution of a normalized sequence of elements of a fixed Wiener chaos was equivalent to the convergence of the fourth moment of the elements of this sequence to three. This result, now known as the fourth moment theorem, was proved to hold as well for sequences of vectors of multiple integrals possibly of different orders by Peccati and Tudor [9] shortly after. On top of the multidimensional central limit theorem, they discovered that for sequences of fixed order chaos elements, componentwise convergence to the Gaussian distribution always implies joint convergence. More specifically, these properties can be stated in the following way.  Non-commutative counterparts of these theorems have been established in the context of the chaos associated with a free Brownian motion {S t : t ≥ 0} in [4] for the one-dimensional limit theorem and in [7] for the multidimensional version for which the authors proved that the property that component-wise convergence implies joint convergence holds as in the classical Brownian case. This result can be stated in the following way. Theorem 1.2 (Nourdin,Peccati and Speicher [7], 2013). Let d ≥ 2 and n 1 , . . . , n d be integers, and consider a positive definite symmetric matrix c = {c(i, j) : i, j = 1, . . . , d}. Let (s 1 , . . . , s d ) be a semicircular family with covariance c (see Definition 2.14). For each i = 1, . . . , d, we consider a sequence f Then, the following three assertions are equivalent, as k → ∞.
(ii) For every i = 1, . . . , d, the random variable I S ni f (i) k converges in distribution to These results, both in the classical as well as in the free case, provide a dramatic simplification to the more conventional method of moments and cumulants in the sense that it is enough to control the second and the fourth moments of the components of a sequence of vector-valued multiple integrals with respect to a classical or free Brownian motion to ensure convergence to a central limit. These theorems have led to a wide collection of new results and inspired several new research directions -see the following constantly updated webpage that lists all the results directly connected to the fourth moment theorems or the techniques developed in their proofs. https://sites.google.com/site/malliavinstein/home In the classical probability setting, the quest for similar results and properties in the framework of the chaoses associated to a Poisson random measure have lead to a wide range of results, both theoretical as well as applied to fields as diverse as stochastic geometry (see e.g. [2,5,11,12,13,14]) or cosmological statistics (see e.g. [1]). Recently, Bourguin and Peccati proved in [3] that a fourth moment theorem holds on the chaos associated with a free Poisson random measure. In view of this result, a natural question is to assess whether or not a multidimensional version of this fourth moment theorem can ECP 21 (2016), paper 55. be established and if the property that component-wise convergence to the semicircular distribution always implies joint convergence holds as is the case on the Brownian and free Brownian chaoses.
This note provides a positive answer to this question (see Theorem 1.3), hence completing the line of chaotic multidimensional limit theorems initiated in [9] and further developed in [10,7].
Then, the following three statements are equivalent.
converges in distribution to (s 1 , . . . , s d ).  The remainder of the paper is organized as follows. Section 2 contains the most relevant elements of free probability theory needed in order to make this note as selfcontained as possible. Theorem 1.3 is proved in Section 3 while Section 4 gathers technical results used in the proof of Theorem 1.3.

Relevant elements of free probability
This section lists the most relevant elements of free probability theory used in the present note. For more details on the tools and notions used below, the reader is referred to the references [3] and [6].

Free probability, free Poisson process and stochastic integrals
Let (A , ϕ) be a tracial W * -probability space, that is A is a von Neumann algebra with involution * and ϕ : A → C is a unital linear functional assumed to be weakly continuous, positive (meaning that ϕ (X) ≥ 0 whenever X is a non-negative element of A ), faithful (meaning that ϕ (XX * ) = 0 ⇒ X = 0 for every X ∈ A ) and tracial (meaning that ϕ (XY ) = ϕ (Y X) for all X, Y ∈ A ). The self-adjoint elements of A will be referred as random variables. Given a random variable X ∈ A , the law of X is defined, as in [6,Proposition 3.13], to be the unique Borel measure on R having the same moments as X. The non-commutative space L 2 (A , ϕ) denotes the completion of A with respect to the norm X 2 = ϕ (XX * ).

Definition 2.2.
The centered semicircular distribution with variance t > 0, denoted by S(0, t), is the probability distribution given by The symmetric aspect of this distribution around zero guaranties that all its odd moments are zero. Furthermore, it is straightforward to check that the even moments are given, Definition 2.3. The free Poisson distribution with rate λ > 0, denoted by P (λ), is the probability distribution defined as follows: (i) if λ ∈ (0, 1], then P (λ) = (1 − λ)δ 0 + λ ν, and (ii) if λ > 1, then P (λ) = ν, where δ 0 stands for the Dirac mass at 0. Here, Definition 2.4. A free Poisson process N consists of: (i) a filtration {A t : t ≥ 0} of von Neumann sub-algebras of A (in particular, A s ⊂ A t for 0 ≤ s < t), (ii) a collection N = {N t : t ≥ 0} of self-adjoint operators in A + (A + denotes the cone of positive operators in A ) such that: (a) N 0 = 0 and N t ∈ A t for all t ≥ 0, (b) for all t ≥ 0, N t has a free Poisson distribution with rate t, and (c) for all 0 ≤ u < t, the increment N t − N u is free with respect to A u , and has a free Poisson distribution with rate t − u.N will denote the collection of random variablesN = N t = N t − t1 : t ≥ 0 , where 1 stands for the unit of A .N will be referred to as a compensated free Poisson process.
For every integer n ≥ 1, the space L 2 R n + ; C = L 2 R n + denotes the collection of all complex-valued functions on R n + that are square-integrable with respect to the Lebesgue measure on R n + .
Definition 2.5. Let n be a natural number and let f be a function in L 2 R n + .
for almost all (t 1 , . . . , t n ) ∈ R n + with respect to the product Lebesgue measure.
3. The function f is called fully symmetric if it is real-valued and, for any permutation σ in the symmetric group S n , f (t 1 , . . . , t n ) = f t σ(1) , . . . , t σ(n) for almost all (t 1 , . . . , t n ) ∈ R n + with respect to the product Lebesgue measure.
ECP 21 (2016), paper 55. Definition 2.6. Let n, m be natural numbers and let f ∈ L 2 R n + and g ∈ L 2 R m + .
Let p ≤ n ∧ m be a natural number. The p-th arc contraction f p g of f and g is the function defined by nested integration of the middle p variables in f ⊗ g: In the case where p = 0, the function f 0 g is just given by f ⊗ g. Similarly, the p-th star contraction f p−1 p g of f and g is the L 2 R n+m−2p+1 + function defined by nested integration of the middle p − 1 variables and identification of the first non-integrated variable in f ⊗ g: , n ≥ 0} is a unital * -algebra, with product rule given, for any n, m ≥ 1, f ∈ L 2 R n + , g ∈ L 2 R m Definition 2.8. Let n ≥ 1 be an integer. We say that a sequence {f k : k ≥ 1} ⊂ L 2 (R n + ) is tamed if the following conditions hold: every f k is bounded and has bounded support and, for every p ≥ 2 and every π ∈ P p j=1 n , the numerical sequence Remark 2.9. As pointed out in [3,Lemma 4.2], there exists sufficient conditions in order for a sequence {f k : k ≥ 1} to be tamed. It basically consists in requiring that {f k : k ≥ 1} concentrates asymptotically, without exploding, around a hyperdiagonal: fix n ≥ 2, and consider a sequence {f k : k ≥ 1} ⊂ L 2 (R n + ). Assume that there exist strictly positive numerical sequences {M k , z k , α k : k ≥ 1} such that α k /z k → 0 as k → ∞ and the following properties are satisfied: (a) the support of f k is contained in the set n j=1 (−z k , z k ), (b) |f k | ≤ M k , (c) f k (t 1 , . . . , t n ) = 0, whenever there exist t i , t j such that |t i − t j | > α k and (d) for every integer p ≥ n, the mapping k → M p k z k α p−1 k is bounded.
Then, the sequence {f k : k ≥ 1} is tamed. and by convention, we order them by their least elements, i.e., min B i < min B j if and only if i < j. The cardinality of a block B is denoted by |B|. A block is said to be a singleton if it has cardinality one. A partition with only blocks of cardinality two is called a pairing. The set of all partitions of [n] is denoted P(n), the set of all pairings is denoted P 2 (n), the set of all partitions without singletons is denoted P ≥2 (n) and the set of all partitions without singletons and with at least one block of cardinality greater or equal to three is denoted P ≥2+ (n). Observe that it holds that P 2 (n) ⊂ P ≥2 (n) ⊂ P(n) and P 2 (n) P ≥2+ (n) = P ≥2 (n). The number of blocks of a partition π ∈ P(n) is denoted by |π|. Definition 2.10. Let π ∈ P(n) be a partition of [n]. π is said to have a crossing if there are two distinct blocks B 1 , B 2 in π with elements x 1 , y 1 ∈ B 1 and x 2 , y 2 ∈ B 2 such that x 1 < x 2 < y 1 < y 2 . If π ∈ P(n) has no crossings, it is said to be a noncrossing partition. The set of non-crossing partitions of [n] is denoted N C(n). The noncrossing elements of P 2 (n), P ≥2 (n) and P ≥2+ (n) are denoted respectively by N C ≥2+ (n), N C ≥2+ (n) and N C ≥2+ (n). In that case too, it holds that N C 2 (n) ⊂ N C ≥2 (n) ⊂ N C ( n) and N C 2 (n) N C ≥2+ (n) = N C ≥2 (n). Definition 2.11. Let n 1 , . . . , n r be positive integers with n = n 1 +· · ·+n r and partition the set [n] according to these integers by putting [n] = B 1 · · · B r , where B 1 = {1, . . . , n 1 }, B 2 = {n 1 + 1, . . . , n 1 + n 2 } and so forth through B r = {n 1 + · · · + n r−1 + 1, . . . , n 1 + · · · + n r } .

Non-crossing partitions, partition integrals and semicircular families
Denote this partition by n 1 ⊗· · ·⊗n r . A partition π ∈ P ≥2 (n) is said to respect n 1 ⊗· · ·⊗n r if no block of π contains more than one element from any given block of n 1 ⊗ · · · ⊗ n r . For any given subset Q(n) ⊂ P(n), the subset of Q(n) consisting of all partitions that respect n 1 ⊗ · · · ⊗ n r is denoted Q (n 1 ⊗ · · · ⊗ n r ). Definition 2.12. Let n 1 , . . . , n r be positive integers and let π ∈ P ≥2 (n 1 ⊗ · · · ⊗ n r ). Let B 1 , B 2 be two blocks in n 1 ⊗ · · · ⊗ n r . π is said to link b 1 and B 2 if there is a block of π containing an element of B 1 and an element of B 2 . Define a graph C π whose vertices are the blocks of n 1 ⊗ · · · ⊗ n r ; C π has an edge between B 1 and B 2 if and only if π links B 1 and B 2 . The partition π on which the graph is based will be said to be connected with respect to n 1 ⊗ · · · ⊗ n r (or that π connects the blocks of n 1 ⊗ · · · ⊗ n r ) if the graph C π is connected. The set N C c ≥2 (n 1 ⊗ · · · ⊗ n r ) will denote the subset of all the partitions in N C c ≥2 (n 1 ⊗ · · · ⊗ n r ) that both respect and connect n 1 ⊗ · · · ⊗ n r . Similarly, the set N C c ≥2+ (n 1 ⊗ · · · ⊗ n r ) will denote the subset of all the partitions in N C ≥2+ (n 1 ⊗ · · · ⊗ n r ) that both respect and connect n 1 ⊗ · · · ⊗ n r . Definition 2.13. Let n ≥ 2 be an integer and let π ∈ P ≥2 (n). Let f : R n + → C be a measurable function. The partition integral of f with respect to π, denoted π f , is defined, when it exists, to be the constant The previous relation implies in particular that, for every i = 1, . . . , d, the random variable s i has a semicircular distribution with mean zero and variance c(i, i).

Proof of Theorem 1.3
Observe that the equivalence between (ii) and (iii) is a direct consequence of [3,Theorem 4.3]. As it is clear that (i) implies (iii), we are left with proving that (iii) implies (i). Assume that (iii) holds and recall that by [3,Theorem 4.3], this is equivalent to the fact that, for each i = 1, . . . , d, for all ∈ {1, . . . , n i − 1} and for all q ∈ {0, . . . , n i − 1}, we Observe that by a straightforward application of Fubini's Theorem along with the mirrorsymmetry of the functions f , so that we also have that, for each i = 1, . . . , d, In order to show (i), we have to show that any moment in the variables converges, as k goes to infinity, to the corresponding moment of the semicircular family (s 1 , . . . , s d ). Let r ≥ 1 and i 1 , . . . , i r be positive integers. Consider the moments The goal is to prove that these moments converge, as k goes to infinity, to ϕ [s i1 · · · s ir ].
By Proposition 4.1, we have We can decompose the sum appearing on the right hand side by isolating the pairings and the rest of the partitions in N C ≥2 (n i1 ⊗ · · · ⊗ n ir ), so that In view of (3.1), the argument used in [7, Proof of Theorem 1.3] guaranties that which is exactly the moment ϕ (s i1 , . . . , s ir ) of a semicircular family with covariance matrix c. Therefore, in order to conclude the proof, it only remains to prove that As pointed out in [4, Remark 1.33] in the case of pairings, it is always possible to decompose a given partition π ∈ N C ≥2+ (n i1 ⊗ · · · ⊗ n ir ) into a disjoint union of connected ECP 21 (2016), paper 55.

≥2
j∈Iq n ij and where I 1 · · · I m is a partition of the set {1, . . . , r}. Hence, the partition integral appearing in (3.2) takes the form As π ∈ N C ≥2+ (n i1 ⊗ · · · ⊗ n ir ), π contains at least one block V * of size greater or equal to three. Assume that this block V * belongs to the connected partition π q * for a certain q * ∈ {1, . . . , m}. This implies that π q * ∈ N C c ≥2 j∈I q * n ij where |I q * | ≥ 3. Hence, using (3.1) along with Proposition 4.2, we get that which, in view of (3.3), concludes the proof.

Auxiliary results
A straightforward generalization of [3,Theorem 3.15] yields the following diagram formula for free Poisson multiple integrals.  Then, for any r ≥ 3 and any partition π ∈ N C c ≥2 (n i1 ⊗ · · · ⊗ n ir ), it holds that Proof. As in Definition 2.11, we denote by B 1 , . . . , B r the blocks of the partition n i1 ⊗ · · · ⊗ n ir . Furthermore, we denote by b i j the elements of the block B i ∈ n i1 ⊗ · · · ⊗ n ir , 1 ≤ i ≤ r and 1 ≤ j ≤ n i . Remark 3.10 in [3], along with the fact that π respects n i1 ⊗ · · · ⊗ n ir , ensures that π contains at least one block V of the form V = b s ns , b s+1 1 for some s ∈ {1, . . . , r − 1}. Note that according to the terminology introduced in Definition 2.12, the block V ∈ π links the blocks B s and B s+1 of n i1 ⊗ · · · ⊗ n ir . Denote by the number of blocks of π of size two (including V ) linking B s and B s+1 . Similarly, denote by p the number of blocks of size strictly greater than two linking B s and B s+1 . Note that, as π is non-crossing, it is necessarily the case that p ∈ {0, 1}. If = 1, there is only one block of size two (namely V ) linking B s and B s+1 . If > 1, then as π is non-crossing and does not contain any singleton, the blocks of size exactly Observe that, in order for the above blocks to make sense, it necessarily holds that 1 ≤ ≤ n s ∧ n s+1 in the case where p = 0 and 1 ≤ ≤ n s ∧ n s+1 − 1 in the case where ECP 21 (2016), paper 55. p = 1. Additionally, the fact that π connects n i1 ⊗ · · · ⊗ n ir excludes the case where p = 0, n s = n s+1 and = n s . This implies that is necessarily such that 1 ≤ ≤ n s ∧ n s+1 − δ ns+1 ns if p = 0 and 1 ≤ ≤ n s ∧ n s+1 − 1 if p = 1 (where δ ns+1 ns denotes the Kronecker delta between n s and n s+1 ). Furthermore, observe that the number of blocks of π linking B s to other blocks of n i1 ⊗ · · · ⊗ n ir but not to B s+1 is given by n s − − p. Similarly, the number of blocks of π linking B s+1 to other blocks of n i1 ⊗ · · · ⊗ n ir but not to B s is given by n s+1 − − p.
Having now a clear view of how the blocks B s and B s+1 can be linked together and how they can be linked to the other blocks of n i1 ⊗ · · · ⊗ n ir allows to specify further the form of the partition function f in |π| variables obtained by identifying the variables t i and t j in the argument of the tensor if and only if i and j are in the same block of π. More specifically, we can write, for any , p as above, where G π, ,p k denotes a function of |π| − variables and where the dot in the arguments and G π, ,p k stands for the remaining |π| − n s − n s+1 + 2p + − 1 variables. Observe that, using this decomposition along with the definition of star contractions given in Definition 2.6, it holds that so that, recalling the definition of partition integrals given in Definition 2.13, Observe that the tameness of the sequences f (i) . (4.2) The last step of this proof will be to show that the contractions appearing in (4.2) are always well defined and that the right-hand side of (4.2) always converges to zero as k goes to infinity in view of the given assumptions. Begin by considering the case where p = 0 and n s = n s+1 . In this case, it holds that 1 ≤ ≤ n s −1 and hence 1 ≤ n s − ≤ n s −1.
This implies that all the contractions appearing in (4.2) are well defined and that, in view of (4.1), it holds that In the case where p = 0, n s = n s+1 and = n s ∧ n s+1 , assume without loss of generality that n s < n s+1 . This yields n s − = 0 and 1 ≤ n s+1 − = n s+1 − n s ≤ n s+1 − 1. This implies that all the contractions appearing in (4.2) are well defined and that, in view of (4.1) and the tameness of the sequence f (is) k Finally, in the case where p = 1, it is always the case that 1 ≤ ≤ n s ∧ n s+1 − 1 so that 0 ≤ n s −(n s ∧ n s+1 ) ≤ n s − −1 ≤ n s −2 and 0 ≤ n s+1 −(n s ∧ n s+1 ) ≤ n s+1 − −1 ≤ n s+1 −2.
This implies that all the contractions appearing in (4.2) are well defined and that, in view of (4.1), it holds that