Convergence of the Fourth Moment and Infinite Divisibility

In this note we prove that, for infinitely divisible laws, convergence of the fourth moment to 3 is sufficient to ensure convergence in law to the Gaussian distribution. Our results include infinitely divisible measures with respect to classical, free, Boolean and monotone convolution. A similar criterion is proved for compound Poissons with jump distribution supported on a finite number of atoms. In particular, this generalizes recent results of Nourdin and Poly.


Introduction and Statement of Results
In a seminal paper, Nualart and Peccati [20] proved a convergence criterion for multiple integrals in a fixed chaos with respect to the classical Brownian motion to the standard normal distribution N (0, 1) which gives a drastic simplification for the so-called method of moments. More precisely, let (W t ) t≥0 be a standard Brownian motion. For every square-integrable function f on R m + we denote by I W m (f ) the m-th multiple Wiener-Itô stochastic integral of f with respect to W . (1) E[X 4 n ] → 3 (2) µ Xn → N (0, 1) In free probability, the standard semicircle distribution S(0, 1) plays the role of the gaussian distribution. Recently, it was proved by Kemp et al. [13] that the Nualart-Peccati criterion also holds for the free Brownian motion (S t ) t≥0 and its multiple Wigner integrals I S m (f ). Theorem 1.2 ( [13]). Let {X n := I S m (f n )} n>0 be a sequence of multiple Wigner integrals in a fixed m-chaos with E[X 2 n ] → 1 denote µ Xn the distribution of X n . Then the following are equivalent (1) E[X 4 n ] → 2 (2) µ Xn → S(0, 1) In this paper we prove analogous results to Theorem 1.1 and Theorem 1.2 in the setting of infinitely divisible laws. Let ID( * ) and ID(⊞) denote the classes of probability measures which are infinitely divisible with respect to classical convolution * and free convolution ⊞, respectively. n ] → 2 then µ Xn → S(0, 1). To complete the picture we show that the monotone probability of Muraki also fits in our framework, namely, Theorems 1.3 and 1.4 are also true in the monotone case.
Theorem 1.5. Let {µ n = µ Xn } n>0 be a sequence of ⊲-infinitely divisible probability measures with common variance 1 and mean 0 and denote by A(0, 1) the arcsine distribution with mean zero and variance 1. If E[X 4 n ] → 1.5 then µ Xn → A(0, 1). Moreover, we can extend our results to compound Poisson distributions whose Lévy measure has finite support. We only state the free version for the sake of clarity.
Theorem 1.6. Let µ Xn ∈ ID(⊞) be random variables and denote by π ⊞ (λ, ν) the free compound poisson measure with rate λ and jump distribution ν . We want to emphasize that our approach relies on a third notion of non commutative independence (Boolean independence) and the so-called Bercovici-Pata bijections, Λ, Λ ⊲ and B (see Section 2). This gives another example on how this third notion of independence sometimes regarded as uninteresting because of its simplicity can provide a better understanding in other notions of independence.
Some natural questions arise from the theorems above. What is the relation between Theorems 1.1 and 1.2, and Theorems 1.3 and 1.4? Multiple integrals are in general not infinitely divisible 1 . However, this is true in the first or second chaos. In particular, from Theorem 1.6 we may recover Theorem 4.3 in Nourdin and Poly [19]. Another interesting question coming from Theorem 1.5 is if Theorem 1.1 is also valid for multiple integrals with respect to monotone Brownian motion; to the knowledge of the author this is still an open question.

Preliminaries
2.1. The Cauchy Transfom. We denote by M the set of Borel probability measures on R. The upper half-plane and the lower half-plane are respectively denoted as C + and C − .
z−x (z ∈ C + ) be the Cauchy transform of µ ∈ M. The relation between weak convergence and the Cauchy Transform is the following (see e.g. [2]). Proposition 2.1. Let µ 1 and µ 2 be two probability measures on R and Then d is a distance which defines a metric for the weak topology of probability measures. In particular, G µ (z) is bounded in {z : ℑ(z) ≥ 1}. 1 A counterexample for the free case is given by the third Chebychev polynomial of a semicircle, i.e., x = s 3 − 2s.
In other words, a sequence of probability measures {µ n } n≥1 on R converges weakly to a probability measure µ on R if and only if for all z with ℑ(z) ≥ 1 we have lim n→∞ G µn (z) = G µ (z).
2.2. The Jacobi Parameters. Let µ be a probability measure with all the moments. The Jacobi parameters where the polynomials P −1 (x) = 0, P 0 (x) = 1 and (P m ) m≥0 is a sequence of orthogonal monic polynomials with respect to µ, that is, A measure µ is supported on m points iff γ m−1 = 0 and γ n > 0 for n = 0, . . . , m−2.
The Cauchy transform may be expressed as a continued fraction in terms of the Jacobi parameters, as follows.
In the case when µ has 2n + 2-moments we can still make an orthogonalization procedure until the level n. In this case the Cauchy transform has the form where ν is a probability measure.

Different notions of convolution.
In non-commutative probability, there exist various notions of independence. In this paper we will focus on the notions of independence coming from universal products as classified by Muraki [16]: tensor(classical), free, Boolean and monotone independence.

Classical convolution.
Recall that the classical convolution of two probability measures µ 1 , µ 2 on R is defined as the probability measure Classical convolution corresponds to the sum of tensor independent random variables: µ a * µ b = µ a+b , for a and b independent random variables. The (classical) cumulants are the coefficients c n = c n (µ) in the series expansion Similar convolutions and related transforms exist for the free, Boolean and monotone theories.

Free convolution.
Free convolution was defined in [23] for compactly supported probability measures and later extended in [14] for the case of finite variance, and in [6] for the general unbounded case. Let G µ (z) be the Cauchy transform of µ ∈ M and let F µ (z) be its reciprocal 1 Gµ(z) . It was proved in Bercovici and Voiculescu [6] that there are positive numbers η and M such that F µ has a right inverse Free additive convolution corresponds to the sum of free random variables: µ a ⊞µ b = µ a+b , for a and b free random variables. The free cumulants [21] are the coefficients κ n = κ n (µ) in the series expansion

Boolean convolution.
The Boolean convolution [22] of two probability measures µ 1 , µ 2 on R is defined as the probability measure Boolean convolution corresponds to the sum of Boolean-independent random variables. Boolean cumulants are defined as the coefficients r n = r n (µ) in the series r n z 1−n .

Monotone convolution.
The monotone convolution was defined in [15] and extended to unbounded measures in [10]. The monotone convolution of two probability measures µ 1 , µ 2 on R is defined as the probability measure µ 1 ⊲ µ 2 on R such that F µ1⊲µ2 (z) = F µ1 (F µ2 (z)), z ∈ C + . Monotone convolution corresponds to the sum of monotone independent random variables. Recently, Hasebe and Saigo [12] have defined a notion of monotone cumulants (h n ) n≥1 which satisfy that h n (µ ⊲k ) = kh n (µ).

Moment-Cumulant formulae.
For a measure µ its classical, free, Boolean and monotone cumulants (c n ) n≥1 , (k n ) n≥1 , (r n ) n≥1 , (h n ) n≥1 , satisfy the momentcumulant formulas where, for a sequence of complex numbers (f n ) n≥1 and a partition π = {V 1 , . . . , V i }, we define f π := f |V1| · · · f |Vi| and |π| is the number of blocks of the partition π and where P(n), N C(n), I(n), M(n) denote the set of all, non-crossing, interval and monotone partitions (see [12,21,22]), respectively. We note here that convergence of the first n moments is equivalent to the convergence of the first n cumulants.
A probability measure µ is said to be ⊛-infinitely divisible if for each n ∈ N there exist µ n ∈ M such that µ = µ ⊛n n . We will denote by ID(⊛) the set of ⊛-infinitely divisible measures.
Recall that a probability measure µ is infinitely divisible in the classical sense if and only if its classical cumulant transform log µ has the Lévy-Khintchine representation where γ ∈ R and σ is a finite measure on R. If this representation exists, the pair (γ, σ) is determined in a unique way and is called the (classical) generating pair of µ. In this case we denote µ by ρ γ,σ * From the Voiculescu Transform one has a representation analogous to Lévy-Kintchine´s. Bercovici and Voiculescu [6] proved that a probability measure µ is ⊞-infinitely divisible if and only if there exists a finite measure σ on R and a real constant γ such that The pair (γ, σ) is called the free generating pair of µ and we denote µ by ρ γ,σ ⊞ . For the Boolean case, things are easier. As shown by Speicher and Wourodi [22], any probability measure is infinitely divisible with respect to the Boolean convolution and there is also a Boolean Lévy-Kintchine representation. Indeed, it follows then by general Nevanlinna-Pick theory that for any probability measure µ there exists a real constant γ and a finite measure σ on R, such that The pair (γ, σ) is called the Boolean generating pair of µ and we denote µ by ρ γ,σ ⊎ . A characterization of ⊲-infinitely divisible measures was done by Muraki [17] and Belinschi [4]. A probability measure µ belongs to ID(⊲) if and only if there exists a composition semigroup of reciprocal Cauchy transforms F s+t = F s • F t = F t • F s and F 1 = F µ . In this case the map t → F t (z) is differentiable for each fixed z in R and we define the mapping A µ on C + by For mappings of this form there exists γ ∈ R and a finite measure σ, such that This is the Lévy-Khintchine formula for monotone convolution and in this case we denote µ by ρ γ,σ ⊲ . The monotone cumulants h n are the coefficients in the series An important class of infinitely divisible measures is the class of compound Poisson distributions since any infinitely divisible measure can be approximated by them.

2.5.
Bercovici-Pata bijections. From the various Lévy-Kintchine representations it is readily seen that there is a bijective correspondence between the infinite divisible measures with respect to the different notions of independence. These bijections are called the Bercovici-Pata bijections, since they were studied by Bercovici and Pata [5] in relation to limit theorems and domain of attractions.

Definition 2.4.
(1) The (classical-to-free) Bercovici-Pata bijection Λ : is defined by the application ρ γ,σ * → ρ γ,σ ⊲ . The weak continuity of Λ and Λ −1 was proved in [3]. On the other hand, the weak continuity of B and B −1 follows from the continuity of the free and Boolean convolution powers since B(µ) = (µ ⊞2 ) ⊎1/2 . Finally the weak continuity of Λ ⊲ was proved in Hasebe [11]. In summary, the arrows of the following commutative diagram are weakly continuous.

Convergence to the Gaussian distribution
For a random variable X with all the moments, mean 0 and variance 1, let us denote by S * n (X) = (X 1 + X 2 + · · · + X n )/ √ n the normalized sum of n independent copies of X. The so-called Central Limit Theorem states that S * n (X) converges, as n → ∞, to the standard Normal distribution N (0, 1).
On the other hand, the free Central Limit Theorem (see [8], [23]) states that the normalized sum of free copies of X converges weakly to the standard semicircle distribution S(0, 1), with density 1 2π Similarly, the Boolean Central Limit Theorem ( [22]) states that the normalized sum of Boolean-independent copies of X converges weakly to the Bernoulli distribution, b := 1/2δ −1 + 1/2δ 1 .
For monotone independence, the limiting distribution for the Central Limit Theorem ( [15]) is the Arcsine distribution, with density The standard proof for convergence to any of these "gaussian" distributions consists in showing the convergence of all the moments. In this section we will prove that, when staying among infinitely divisible laws, convergence of the 4th moment is enough, namely we will prove Theorems 1.3, 1.4 and 1.5.
The main observation is that the following simple lemma together with the continuity properties of the Bercovici-Pata bijections gives the desired results.  n ] → 1.5 then µ Xn → A(0, 1). Proof. We first prove (2). Recall from Definition 2.4 that B stands for the Booleanto-free Bercovici-Pata bijection. Assume µ n = B(ν n ), for some ν n . Then by Remark 2.5, ν n has variance 1 and mean 0, and m 4 (ν n ) → 1. The previous lemma applies and yields that ν n → b. By the continuity of B, we deduce that B(ν n ) → B(b) = S(0, 1). Parts (1) and (3)   . Let X i be Boolean independent identically distributed random variables with E(X i ) = 0 and E(X 2 i ) = 1. Then, the random variable Y n = X1+X2+...Xn √ n is infinitely divisible. Moreover, E(Y n ) = 0, E(Y 2 n ) = 1 and E(Y 4 n ) = r 2 + r 4 /n → r 2 = 1. Thus, by Lemma 3.1, we see that Example 3.4 (Convergence of the Poisson Distribution to the Normal Distribution). Let X n be a random variable with distribution π(n, δ 1 ), the random variable Y n = Xn−n √ n converges weakly to N (0, 1). Indeed, Y n = Xn−n √ n is infinitely divisible. Moreover, E(Y n ) = 0, E(Y 2 n ) = 1 and E(Y 4 n ) = c 4 /n + 3c 2 2 = 1/n + 3 → 3. Hence, by Theorem 3.2 we see that Y n → N (0, 1) A similar argument proves the following criteria for approximation to the Poisson distributions. This shall be compared to the results in [18].
As an example we get the analog of Theorem 1.1 in Deya and Nourdin [9], for the so-called Tetilla Law.
This is the distribution of free commutator s 1 s 2 + s 2 s 1 where s 1 and s 2 are free semicircle variables. If X n ∈ ID(⊞) is a sequence of random variables such that E[X i n ] → m i (µ), for i = 1, ..., 6, then X n converges in distribution to T . Indeed, T is a free compound Poisson π ⊞ (λ, ν) with λ = 2 and ν = b, thus by Theorem 1.5 we get the desired result.
Proof. The fact the (i) implies (ii) follows from the boundedness of the sequence. To prove that (ii) implies (i) we note that since F n is in the first and second chaos then F n is freely infinitely divisible. Moreover F has a Lévy pair (γ, σ) with σ with finite support. We note here that (i-a) corresponds to the Gaussian part.