Ivan Nourdin
email: inourdin@gmail.com

Guillaume Poly
email: guillaume.poly@crans.org

Convergence in law in the second Wiener/Wigner chaos

Keywords: Convergence in law, second Wiener chaos, second Wigner chaos, quadratic form, free probability. 2000 Mathematics Subject Classification: 46L54, 60F05, 60G15, 60H05

, in which Nualart and Peccati discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we offer analogues of this result in the case of free Brownian motion and double Wigner integrals, in the context of free probability.

Introduction

Let X be a centered Gaussian process defined on, say, the time interval [0, 1]. Consider the sequence of its quadratic variations

V n = n-1 k=0 X (k+1)/n -X k/n 2 , n 1.
Very often (e.g. when one seeks to build a stochastic calculus with respect to X, or when one wants to estimate an unknown parameter characterizing X) one is interested in determining, provided it exists, the limit in law of V n properly normalized, namely,

F n = V n -E[V n ] σ n , n 1, with σ n = Var(V n ). (1.1)
For instance, when X is a fractional Brownian motion of Hurst parameter H ∈ (0, 1), it is known that F n is asymptotically normal when H ∈ (0, 3/4], whereas it converges to the Rosenblatt random variable when H ∈ (3/4, 1). Applying the Gram-Schmidt process leads to the existence of a family {e k,n } 0 k n of L 2 (R + ) satisfying ∞ 0 e k,n (x)e l,n (x)dx = E X (k+1)/n -X k/n X (l+1)/n -X l/n , 0 k, l n -1, Fix an integer q 1. In the late 60's, Schreiber [START_REF] Schreiber | Fermeture en probabilité de certains sous-espaces d'un espace L 2[END_REF] proved that the qth Wiener chaos associated with W is closed under convergence in probability. In the present paper, we are rather interested in convergence in law. More precisely, we aim to answer the following question: "Can we describe all the limits in law associated with sequences in the qth Wiener chaos?"

(1.

2)

The answer to (1.2) turns out to be trivial when q = 1: it is indeed well-known (and easy to prove) that, if a sequence of centered Gaussian random variables converge in law, its limit is centered and Gaussian as well. In contrast, to provide an answer to (1.2) when q 3 is a difficult problem which is not yet solved. When q = 2, the answer to (1.2) is almost contained in reference [START_REF] Sevastyanov | A class of limit distributions for quadratic forms of normal stochastic variables[END_REF] by Sevastyanov. Therein, the author considers quadratic forms of the form

F n = n i,j=1 a ij (N i N j -E[N i N j ]), (1.3)
where N 1 , N 2 , . . . ∼ N (0, 1) are independent, and characterizes all the possible shapes of any limit in law of a sequence of the form (1.3). On the other hand, it is a classical result that any element F in the second Wiener chaos can be suitably decomposed as

F (law) = ∞ k=1 λ k (N 2 k -1), (1.4)
with N 1 , N 2 , . . . ∼ N (0, 1) independent and {λ k } k 1 ⊂ R satisfying k λ 2 k < ∞. Thus, sequences in the second Wiener chaos are almost all of the form (1.3) and, as a result, the work [START_REF] Sevastyanov | A class of limit distributions for quadratic forms of normal stochastic variables[END_REF] is not far away to give a complete answer to our question (1.2) in the case q = 2. In the present paper, we solve (1.2) in full generality for q = 2, by using another route compared to [START_REF] Sevastyanov | A class of limit distributions for quadratic forms of normal stochastic variables[END_REF]. Our approach consists to observe that the identity in law (1.4) leads to an expression for the characteristic function φ of F , from which it follows that 1/φ 2 lies in the Laguerre-Pólya class consisting of all real entire functions which are locally uniform limits of real polynomials with real zeros. Then, relying to a lemma from [START_REF] Hellerstein | Limits of entire functions whose growth and zeros are restricted[END_REF], we deduce that the set L of all limiting laws associated to sequences in the second Wiener chaos is composed of those random variables F ∞ of the form

F ∞ law = N + G, (1.5)
where G lies in the second Wiener chaos and N ∼ N (0, λ 2 0 ) is independent of W . Thus, for any centered Gaussian process X, the limit in law of F n given by (1.1) (provided it exists) necessarily has the form (1.5). Another direct consequence of (1.5) is that each random variable in L \ {0} has a law which is absolutely continuous with respect to the Lebesgue measure.

Let us now describe the main result of the present paper. Relying to (1.5), we exhibit a large subset L 0 ⊂ L satisfying that, for any F ∞ ∈ L 0 , the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F ∞ . Theorem 3.4, which contains the precise statement, is in the same spirit than the seminal paper [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF], in which Nualart and Peccati discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. For an overview of the existing literature around this theme, we refer the reader to the book [START_REF] Nourdin | Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality[END_REF], to the survey [START_REF] Nourdin | Lectures on Gaussian approximations with Malliavin calculus[END_REF], and to the constantly updated webpage www.iecn.u-nancy.fr/ nourdin/steinmalliavin.htm.

Finally, the last part of our paper investigates analogues of all the previous results in the free probability setting. We are motivated by the fact that there is often a close correspondence between classical probability and free probability. As one will see, there is no exception to the rule here. Indeed, let S = (S t ) t 0 be a free Brownian motion (defined on a free tracial probability space (A, ϕ)) and let H S 2 denote the second (symmetric) Wigner chaos associated with S, that is, H S 2 is the closed linear subspace of L 2 (ϕ) generated by the family

∞ 0 h(t)dS t 2 - ∞ 0 h 2 (t)dt : h ∈ L 2 (R + ) ,
where ∞ 0 h(t)dS t stands for the Wigner integral of h, in the sense of Biane and Speicher [START_REF] Biane | Stochastic calculus with respect to free Brownian motion and analysis on Wigner space[END_REF]. Then, if F ∞ is the limit in law of a given sequence {F n } belonging to H S 2 , we prove that there exists a random variable A, distributed according to the semicircular law, as well as another random variable G ∈ H S 2 , freely independent of S, such that

F ∞ law = A + G. (1.6)
Then, using (1.6) (which is the exact analogue of (1.5)), we can prove Theorem 4.3, which is the free counterpart of Theorem 3.4.

The rest of this paper is organized as follows. In Section 2, we introduce the notation and we give several preliminary lemmas which will be used to state and prove our main results in the following sections. Section 3 is devoted to the proofs of the results described in the introduction in the case of the standard Brownian motion. Finally, Section 4 deals with the free probability context.

Preliminaries

This section gathers the material we shall need to state and prove the results of this paper.

2.1 Four objects associated to square-integrable symmetric functions of two variables

In the whole paper, we shall deal with symmetric functions of L 2 (R 2 + ) (that is, functions f which are square-integrable and that satisfy f (x, y) = f (y, x) a.e.). We denote by L 2 s (R 2 + ) the set of such functions.

With every function f ∈ L 2 s (R 2 + ), we associate the selfadjoint Hilbert-Schmidt operator

T f : L 2 (R + ) → L 2 (R + ), g → ∞ 0 f (•, y)g(y)dy,
as well as the following four objects which are related to it:

-We write {λ k (f )} k 1 ⊂ R to indicate the eigenvalues of T f and we assume that

|λ 1 (f )| |λ 2 (f )| . . .; -We set E(f ) = {λ k (f )} k 1 \ {0}; -We denote by rank(f ) the rank of T f , that is, r = rank(f ) if and only if λ r (f ) = 0 and λ r+1 (f ) = 0; -We denote by a(f ) the cardinality of E(f ), that is, a(f ) is the number of distinct non-zero eigenvalues of T f .

Second Wiener chaos

Let W = (W t ) t 0 be an ordinary Brownian motion on (Ω, F, P ), and assume that F is generated by W . In this subsection, we focus on elements in the second Wiener chaos, that is, we focus on random variables of the type

F = I W 2 (f ), with f ∈ L 2 s (R 2 +
). For more details on Wiener chaos, we refer to [START_REF] Nourdin | Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality[END_REF]. The following proposition unveils a link between the elements of the second Wiener chaos and some of the objects introduced in the previous subsection.

Proposition 2.1 For any element f ∈ L 2 s (R 2 +
), the following equality holds:

I W 2 (f ) (law) = ∞ k=1 λ k (f ) N 2 k -1 , (2.7)
where {N k } k 1 is a sequence of independent N (0, 1) random variables, and the series converges in L 2 (Ω) and almost surely.

Proof. The proof is standard and omitted. See, e.g., [START_REF] Nourdin | Normal Approximations Using Malliavin Calculus: from Stein's Method to Universality[END_REF]Proposition 2.7.13].

By using (2.7), it is easily seen that the cumulants of F are given by

κ 1 (I W 2 (f )) = 0 and κ r (I W 2 (f )) = 2 r-1 (r -1)! ∞ k=1 λ k (f ) r , r 2.
See, e.g., [9, Identities (2.7.17)].

We shall also need the following hypercontractivity property.

Theorem 2.1 Let F be a double Wiener-Itô integral. Then, for all r 1, we have

E[|F | r ] (r -1) r E[F 2 ] r/2 . (2.8)
Proof. See, e.g., [9, Corollary 2.8.14].

Second Wigner chaos

Let S = (S t ) t 0 be a free Brownian motion, defined on a non-commutative probability space (A, ϕ). That is, S is a stochastic process starting from 0, with freely independent increments, and such that S(t) -S(s) ∼ S(0, ts) is a centered semicircular random variable with variance ts for all t s. (We may think of free Brownian motion as 'infinite-dimensional matrix-valued Brownian motion'.) For more details about the construction and features of S, see [1, Section 1.1] and the references therein.

In this subsection, we focus on elements in the (symmetric) second Wigner chaos, that is, we focus on random variables of the type

F = I S 2 (f ), with f ∈ L 2 s (R 2 + ).
The following result is nothing but the free counterpart of Proposition 2.1.

Proposition 2.2 For any element f ∈ L 2 s (R 2 +
), the following equality in law holds:

I S 2 (f ) law = ∞ k=1 λ k (f ) S 2 k -1 , (2.9)
where {S k } k 1 is a sequence of freely independent centered semicircular random variables with unit variance, and the series converges in L 2 (ϕ).

Proof. It is an immediate extension of [9, Proposition 2.7.13] to the free case.

An immediate consequence of Proposition 2.2 is that the free cumulants (see [START_REF] Nica | Lectures on the Combinatorics of Free Probability[END_REF]) of F are given by

κ 1 (I S 2 (f )) = 0 and κ r (I S 2 (f )) = ∞ k=1 λ k (f ) r , r 2.

Our results in the classical Brownian motion case

Let W = (W t ) t 0 be an ordinary Brownian motion on (Ω, F, P ). The next theorem describes all the limits in law associated with sequences in the second Wiener chaos.

Theorem 3.1 Let {F n } n 1 be a sequence of double Wiener integrals that converges in law to F ∞ . Then, there exists

λ 0 ∈ R and f ∈ L 2 s (R 2 + ) such that F ∞ (law) = N + I W 2 (f ),
where N ∼ N (0, λ 2 0 ) is independent of the underlying Brownian motion W .

During the proof of Theorem 3.1, we shall need the following result taken from [START_REF] Hellerstein | Limits of entire functions whose growth and zeros are restricted[END_REF] (more precisely, it is a suitable combination of Lemma 1 and Lemma 2 in [START_REF] Hellerstein | Limits of entire functions whose growth and zeros are restricted[END_REF]). Lemma 3.2 Let {G n } be a sequence of entire functions of the form

G n (z) = e αnz+βn ∞ k=1 (1 -z/z k,n ) e z/z k,n , z ∈ C, with α n , β n ∈ C,
and where the zeros {z k,n } k 1 of G n are included in R \ {0} and satisfy the condition

∞ k=1 |z k,n | -2 M, n = 1, 2, . . . ,
for some constant M independent of n. Assume that G n converges uniformly on a disc about the origin, to a limit function ≡ 0. Then G n converge uniformly on every bounded set, to an entire function G ∞ of the form

G ∞ (z) = e az 2 +bz+c ∞ k=1 (1 -z/z k ) e z/z k , (3.10)
where a, b, c ∈ C and where the z k are real and such that

∞ k=1 |z k | -2 < ∞.
We shall also need the following result, which is a straightforward consequence of the Paley inequality (3.11) as well as the hypercontractivity property (2.8). Lemma 3.3 Let {F n } n 1 be a tight sequence of double Wiener-Itô integrals. Then

sup n 1 E[|F n | p ] < ∞ for all p 1.
Proof. Let Z be a positive random variable such that E[Z] = 1 and let θ ∈ (0, 1). Consider the decomposition Z = Z1 {Z>θ} + Z1 {Z θ} and take the expectation. One deduces, using Cauchy-Schwarz, that E[Z 2 ] P (Z > θ) + θ 1, that is (Paley inequality),

E[Z 2 ] P (Z > θ) (1 -θ) 2 . ( 3
Z = F 2 n /E[F 2 n ], P F 2 n > θE[F 2 n ] 1 81 (1 -θ) 2 .
(3.12)

The sequence {F n } n 1 being tight, one can choose M > 0 large enough so that P (F 2 n > M )

1 324

for all n 1. By applying (3.12) with θ = 1/2, one gets that

P F 2 n > M 1 324 P F 2 n > 1 2 E[F 2 n ] ,
from which one deduce that

sup n 1 E[F 2 n ] 2M < ∞.
The desired conclusion follows (once again!) from the hypercontractivity property (2.8).

We are now in a position to prove Theorem 3.1.

Proof of Theorem 3.1. We first observe that sup n 1 E[|F n | p ] < ∞ for all p 1 by Lemma 3.3. Hence, without loss of generality, we may and do assume that sup n 1 E[F 2 n ] 1. The rest of the proof is divided into two steps.

Step 1. We claim that φ n , defined on Ω = {z ∈ C : |Rez| < e -1 } as

φ n (z) = E[e zFn ],
converges uniformly on compact sets of Ω, towards an holomorphic function noted φ ∞ . First, observe that the function φ n is well-defined and uniformly bounded on compact sets of Ω. Indeed, by hypercontractivity (2.8) and for every q > 2,

E[|F n | q ] 1/q q -1. Thus, P [|F n | > u]
u -q (q -1) q for every u > 0. Choosing q = q(u) = 1 + u/e, the previous relation shows that P [|F n | > u] e -u/e for every u > e. By a Fubini argument, Since the sequence {φ n } n 1 is bounded on compact sets of Ω, one can assume by Montel's theorem that it converges uniformly on compact sets of Ω (towards, say, h) and one is thus left to show that the limit is unique. Since F n law → F ∞ , we have that φ n (it) converges pointwise to φ ∞ (it) for all t ∈ R (with obvious notation). We deduce that h(it) = φ ∞ (it) for all t ∈ R. But, as an immediate consequence of (3.13) and of the continuous mapping theorem, we have that the function φ ∞ (z) = E[e zF∞ ] is well-defined and holomorphic on Ω. Hence, h = φ on Ω and the desired claim follows.

sup n 1 E[|e zFn |] sup n 1 E[e |Rez| |Fn| ] = sup n 1 1 + |Rez| ∞ 0 e |Rez|u P |F n | > u du < ∞ (3.
Step 2. Let {N k } k 1 denote a sequence of independent N (0, 1) random variables. Since F n = I 2 (f n ) is a double Wiener-Itô integral, its law can be expressed (see Proposition 2.1) as

F n law = ∞ k=1 λ k,n (N 2 k -1)
where the λ k,n are the eigenvalues of the Hilbert-Schmidt operator

T n : L 2 (R + ) → L 2 (R + ) defined as T n (g) = f n , g L 2 (R + ) . In particular, ∞ k=1 λ 2 k,n = 1 2 E[F 2 n ] < ∞. It is straightforward to check that G n (z) := 1 φ n (z) 2 = ∞ k=1 (1 -2λ k,n z) e 2λ k,n z , z ∈ Ω.
Since φ ∞ is not identically zero and φ n (z) = 0 for all z ∈ Ω, the Hurwitz principle applies and yields that φ ∞ (z) = 0 for all z ∈ Ω. Therefore, G n converges uniformly on compact sets of Ω to G ∞ = φ -2 ∞ . Thanks to Lemma 3.2, we deduce that G ∞ has the form

G ∞ (z) = e az 2 +bz+c ∞ k=1 (1 -2λ k z) e 2λ k z , z ∈ Ω, (3.14)
where a, b, c ∈ C and where the λ k are real and such that ∞ k=1 λ 2 k < ∞. In (3.14), we necessarily have a ∈ R + , b = 0 and e c = 1.

Indeed, e c = G ∞ (0) = 1. Moreover, ∞ k=1 (1 -2λ k z) e 2λ k z = e -∞ j=2 (2z) j j ∞ k=1 λ j k , z ∈ Ω, so that b = G ′ ∞ (0) = -2E[F ∞ ] = 0 and 2a -4 ∞ k=1 λ 2 k = G ′′ ∞ (0) = -2E[F 2 ∞ ], implying that a = 2 ∞ k=1 λ 2 k -E[F 2 ∞ ] ∈ R.
Moreover, using Fatou's lemma we deduce that

2 ∞ k=1 λ 2 k 2 lim inf n→∞ ∞ k=1 λ 2 m k n ,n 2 lim inf n→∞ ∞ k=1 λ 2 k,n = lim inf n→∞ E[F 2 n ] = E[F 2 ∞ ].
That is, a 0. Consequently, (3.14) implies that

F ∞ law = N + ∞ k=1 λ k (N 2 k -1), (3.15)
where N ∼ N (0, -a) is independent from N k , k 1. To conclude the proof, it suffices to observe that

∞ k=1 λ k (N 2 k -1) law = I W 2 (f ),
where f is given by f (x, y) = ∞ k=1 e k (x)e k (y), with {e k } k 1 any orthonormal basis of L 2 (R + ).

In the next theorem, we exhibit a large subset L 0 ⊂ L such that, for any F ∞ ∈ L 0 , the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to F ∞ .

Theorem 3.4 Let f ∈ L 2 s (R 2 + ) with 0 rank(f ) < ∞, let µ 0 ∈ R and let N ∼ N (0, µ 2 0 ) be independent of the underlying Brownian motion W . Assume that |µ 0 | + f L 2 (R + ) > 0 and set Q(x) = x 2(1+1 {µ 0 =0} ) a(f ) i=1 (x -λ i (f )) 2 .
Let {F n } n 1 be a sequence of double Wiener-Itô integrals. Then, as n → ∞, we have

(i) F n law → N + I W 2 (f )
if and only if all the following are satisfied:

(ii-a) κ 2 (F n ) → κ 2 (N + I W 2 (f )) = µ 2 0 + 2 f 2 L 2 (R 2 + ) ; (ii-b) degQ r=3 Q (r) (0) r! κr(Fn) (r-1)!2 r-1 → degQ r=3 Q (r) (0) r! κr(I W 2 (f )) (r-1)!2 r-1 ; (ii-c) κ r (F n ) → κ r (I W 2 (f )) for a(f ) consecutive values of r, with r 2(1 + 1 {µ 0 =0} ).
Before doing the proof of Theorem 3.4, let us detail two explicit examples.

1. Consider first the situation where

f L 2 (R 2 + ) = 0 (that is, rank(f ) = a(f ) = 0) and µ 0 = 0. In this case, Q(x) = x 4 and condition (ii -c) is immaterial. Therefore, F n law → N (0, µ 2 0 ) if and only if κ 2 (F n ) → µ 2 0 (condition (ii -a)) and κ 4 (F n ) → 0 (conditions (ii -b))
. As such, one recovers the celebrated Nualart-Peccati criterion (in the case of double integrals), see [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF].

2. Consider now the situation where, in Theorem 3.4, one has r = rank(f ) < ∞, a(f ) = 1, λ 1 (f ) = 1 and µ 0 = 0. This corresponds to the case where the limit I W 2 (f ) is a centered chi-square random variable with r degrees of freedom. In this case,

Q(x) = x 4 -2x 3 + x 2 . Therefore, F n law → I W 2 (f ) if and only if κ 2 (F n ) → 2r (conditions (ii -a) and (ii -c)) and κ 4 (F n ) -12 κ 3 (F n ) → -48r (condition (ii -b))
. As such, one recovers a theorem of Nourdin and Peccati in the special case of double integrals, see [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF].

For the proof of Theorem 3.4, we shall need the following auxiliary lemma. Lemma 3.5 Let µ 0 ∈ R, let a ∈ N * , let µ 1 , . . . , µ a = 0 be pairwise distinct real numbers, and let m 1 , . . . , m a ∈ N * . Set

Q(x) = x 2(1+1 {µ 0 =0} ) a i=1 (x -µ i ) 2
Assume that {λ j } j 0 is a square-integrable sequence of real numbers satisfying

λ 2 0 + ∞ j=1 λ 2 j = µ 2 0 + a i=1 m i µ 2 i (3.16) 2(1+1 {µ 0 =0} +a) r=3 Q (r) (0) r! ∞ j=1 λ r j = 2(1+1 {µ 0 =0} +a) r=3 Q (r) (0) r! a i=1 m i µ r i (3.17) ∞ j=1 λ r j = a i=1 m i µ r i for 'a' consecutive values of r 2(1 + 1 {µ 0 =0} ). (3.18)
Then:

(i) |λ 0 | = |µ 0 |.
(ii) The cardinality of the set S = {j 1 : λ j = 0} is finite.

(iii) {λ j } j∈S = {µ i } 1 i a .

(iv) for any i = 1, . . . , a, one has m i = #{j ∈ S : λ j = µ i }.

Proof. We divide the proof according to the nullity of µ 0 .

First case: µ 0 = 0. We have

Q(x) = x 2 a i=1 (x -µ i ) 2 .
Since the polynomial Q can be rewritten as

Q(x) = 2(1+a) r=2 Q (r) (0) r! x r , assumptions (3.16
) and (3.18) together ensure that

λ 2 0 a i=1 µ 2 i + ∞ j=1 Q(λ j ) = a i=1 m i Q(µ i ) = 0.
Because Q is positive and a i=1 µ 2 i = 0, we deduce that λ 0 = 0 and Q(λ j ) = 0 for all j 1, that is, λ j ∈ {0, µ 1 , . . . , µ a } for all j 1. This shows claims (i) as well as:

{λ j } j∈S ⊂ {µ i } 1 i a . (3.19)
Moreover, since the sequence {λ j } j 1 is square-integrable, claim (ii) holds true as well. It remains to show (iii) and (iv). For any i = 1, . . . , a, let n i = #{j ∈ S : λ j = µ i }. Also, let r 2 be such that r, r + 1, . . . , r + a -1 are 'a' consecutive values satisfying (3.17). We then have

     µ r 1 µ r 2 • • • µ r a µ r+1 1 µ r+1 2 • • • µ r+1 a . . . . . . . . . . . . µ r+a-1 1 µ r+a-1 2 • • • µ r+a-1 a           n 1 -m 1 n 2 -m 2 . . . n a -m a      =      0 0 . . . 0      .
Since µ 1 , . . . , µ a = 0 are pairwise distinct, one has (Vandermonde matrix)

det      µ r 1 µ r 2 • • • µ r a µ r+1 1 µ r+1 2 • • • µ r+1 a . . . . . . . . . . . . µ r+a-1 1 µ r+a-1 2 • • • µ r+a-1 a      = a i=1 µ r i × det      1 1 • • • 1 µ 1 µ 2 • • • µ a . . . . . . . . . . . . µ a-1 1 µ a-1 2 • • • µ a-1 a      = 0,
from which (iv) follows. Finally, recalling the inclusion (3.19) we deduce (iii).

Second case: µ 0 = 0. In this case, one has Q(x) = x 4 a i=1 (xµ i ) 2 and claims (ii), (iii) and (iv) may be shown by following the same line of reasoning as above. We then deduce claim (i) by looking at (3.16).

We are now in a position to prove Theorem 3.4.

Proof of Theorem 3.4. The implications (i) → (ii-a), (ii-b), (ii-c) are immediate consequences of the Continuous Mapping Theorem together with Lemma 3.3. Now, assume (ii-a), (ii-b), (ii-c) and let us show that (i) holds true. The sequence {F n } being bounded in L 2 , it is tight by Prokhorov's theorem. Hence, to prove claim (i) it is sufficient to show that any subsequence {F n ′ } converging in law to some random variable F ∞ is necessarily such that F ∞ law = N + I W 2 (f ). From now on, and only for notational convenience, we assume that {F n } itself converges to F ∞ . By hypercontractivity (2.8), one has that κ r (F n ) → κ r (F ∞ ) for all r. Thanks to Theorem 3.1, we know that

F ∞ law = λ 0 N 0 + ∞ j=1 λ j (N 2 j -1), (3.20)
for some λ 0 , λ 1 , . . . satisfying j λ 2 j < ∞ and where N 0 , N 1 , N 2 , . . . ∼ N (0, 1) are independent. Combining our assumptions (ii-a),(ii-b),(ii-c) with (3.20), we deduce that (3.16)-(3.17)- (3.18) hold true. If a = 0, we deduce from Lemma 3.5 that G ∞ law = F ∞ , thereby concluding the proof of Theorem 3.4. If a = 0 and µ = 0, then the desired conclusion follows from the Nualart-Peccati criterion of asymptotic normality. Finally, if a = 0 and µ = 0, then (3.16) implies that F ∞ = 0 a.s., and the proof of Theorem 3.4 is also concluded in this case. [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF] Our results in the free Brownian motion case Let S = (S t ) t 0 be a free Brownian motion, defined on a non-commutative probability space (A, ϕ). The following result fully describes all the possible limiting laws for sequences in the second (symmetric) Wigner chaos. It is the exact analogue of Theorem 3.1.

Theorem 4.1 Let {F n } n 1 be a sequence in the second Wigner chaos that converges in law to F ∞ . Then, there exists λ

0 ∈ R and f ∈ L 2 s (R 2 + ) such that F ∞ law = A + I S 2 (f ),
where A ∼ S(0, λ 2 0 ) is freely independent of the underlying free Brownian motion S.

Proof. We have

F n = I S 2 (f n ) with f n ∈ L 2 s (R 2 + ) for each n 1. Set G n = I W 2 (f n )
where W = (W t ) t 0 stands for an ordinary Brownian motion. We have

E[G 2 n ] = 2 f n 2 L 2 (R 2 + ) = 2ϕ(F 2 n ) → 2ϕ(F 2 ∞ )
as n → ∞, so {G n } is bounded in L 2 (Ω). By Prokhorov's theorem, it is tight and there exists a subsequence {G n ′ } that converges in law to, say, G ∞ . By Theorem 3.1, the limit G ∞ has necessarily the form

G ∞ = N + I W 2 (f ) with f ∈ L 2 (R 2 + ) and N ∼ N (0, 2λ 2 0 ) independent of W . Using Lemma 3.3, we deduce that, as n ′ → ∞, 2 ∞ k=1 λ k (f n ′ ) 2 = κ 2 (G n ′ ) → κ 2 (G ∞ ) = 2λ 2 0 + 2 ∞ k=1 λ k (f ) 2 and 2 r-1 (r -1)! ∞ k=1 λ k (f n ′ ) r = κ r (G n ′ ) → κ r (G ∞ ) = 2 r-1 (r -1)! ∞ k=1 λ k (f ) r , r 3. Now, since F n converges in law to F ∞ , we have, as n → ∞, ∞ k=1 λ k (f n ) r = κ r (F n ) → κ r (F ∞ ) for any r 2.
We deduce that

κ 2 (F ∞ ) = λ 2 0 + ∞ k=1 λ k (f ) 2 and κ r (F ∞ ) = ∞ k=1 λ k (f ) r , r 3,
from which the announced claim follows.

Using cumulants (as in the previous proof, see also Proposition 3.3), it is straightforward to check the following Wiener-Wigner transfer principle between limits in the second Wiener chaos and limits in the second Wigner chaos. Theorem 4.2 Let {f n } n 1 be a sequence of elements of L 2 s (R 2 + ), let W = (W t ) t 0 be an ordinary Brownian motion and let S = (S t ) t 0 be a free Brownian motion. Let N ∼ N (0, 2λ 2 0 ) (resp. A ∼ S(0, λ 2 0 )) be independent of W (resp. S). Then, as n → ∞,

I W 2 (f n ) law → N + I W 2 (f ) if and only if I S 2 (f n ) law → A + I S 2 (f ).
Also, reasoning as in the proof of Theorem 3.4, we may obtain the following characterization in terms of a finite number of cumulants for convergence in law within the second Wigner chaos.

Theorem 4.3 Let f ∈ L 2 s (R 2 + ) with 0 rank(f ) < ∞, let µ 0 ∈ R and let A ∼ S(0, µ 2
0 ) be independent of the underlying free Brownian motion S. Assume that |µ 0 | + f L 2 (R + ) > 0 and set

Q(x) = x 2(1+1 {µ 0 =0} ) a(f ) i=1 (x -λ i (f )) 2 .
Let {F n } n 1 be a sequence of double Wigner integrals. Then, as n → ∞, we have (ii-c) κ r (F n ) → κ r (I W 2 (f )) for a(f ) consecutive values of r, with r 2(1 + 1 {µ 0 =0} ).

To conclude, we describe some easy consequences of Theorem 4.3:

1. Consider first the situation where f L 2 (R 2 + ) = 0 (that is, rank(f ) = a(f ) = 0) and µ 0 = 0. In this case, Q(x) = x 4 and condition (iic) is immaterial. Therefore, F n law → S(0, µ 2 0 ) if and only if κ 2 (F n ) → µ 2 0 (condition (iia)) and κ 4 (F n ) → 0 (conditions (iib)). As such, one recovers a result by Kemp, Nourdin, Peccati and Speicher (in the case of double integrals), see [START_REF] Kemp | Wigner chaos and the fourth moment[END_REF].

2. Consider now the situation where, in Theorem 3.4, one has r = rank(f ) < ∞, a(f ) = 1, λ 1 (f ) = 1 and µ 0 = 0. This corresponds to the case where the limit I S 2 (f ) is a centered free Poisson random variable with rate r. In this case, Q(x) = x 4 -2x 3 + x 2 . Therefore, F n law → I S 2 (f ) if and only if κ 2 (F n ) → r (conditions (ii-a) and (ii-c)) and κ 4 (F n )-2 κ 3 (F n ) → -r (condition (iib)). As such, one recovers a theorem of Nourdin and Peccati in the special case of double integrals, see [START_REF] Nourdin | Poisson approximations on the free Wigner chaos[END_REF].

3. Finally, consider the situation µ 0 = 0 and f = e 1e 2 , where e 1 , e 2 ∈ L 2 (R + ) are orthogonal and have norm 1 (thus, r = rank(f ) = a(f ) = 2). This corresponds to the case where the limit is S 2 1 -S 2 2 , with S 1 , S 2 ∼ S(0, 1) independent. That is, the limit is distributed according to the tetilla law. In this case, Q(x) = x 6 -2x 4 + x 2 . Therefore, F n law → S 2 1 -S 2 2 if and only if κ 2 (F n ) → 2 (conditions (iia) and (iic)), κ 4 (F n ) → 2 (condition (iic)) and κ 6 (F n ) -2κ 4 (F n ) → -2 (condition (iib)). As such, one recovers a theorem of Deya and Nourdin in the special case of double integrals, see [START_REF] Deya | Convergence of Wigner integrals to the tetilla law[END_REF].

  13) for any z ∈ C such that |Rez| < e -1 . Moreover, since ∂T φ n (z)dz = ∂T E[e zFn ]dz = E ∂T e zFn dz = 0 for all triangle ∂T ⊂ Ω, one deduces from the Goursat theorem that each φ n is holomorphic on Ω.

+ f 2 L 2

 22 (i) F n law → A + I S 2 (f )if and only if all the following are satisfied:(ii-a) κ 2 (F n ) → κ 2 (A + I S 2 (f )) = µ 2 0

Acknowledgments. We are grateful to Giovanni Peccati for bringing to our attention reference [14] and for pointing out an error in a previous version (which is unfortunately the published version [10], see also [11]) of this work.

; IN was partially supported by the ANR grants ANR-09-BLAN-0114 and ANR-10-BLAN-0121.


Marie-Hélène Masson
email: mmas-son@hds.utc.fr

Thierry Denoeux
email: thierry.denoeux@hds.utc.fr

Ranking from pairwise comparisons in the belief functions framework

The problem of deriving a binary relation over alternatives based on paired comparisons is studied. The problem is tackled in the framework of belief functions, which is well-suited to model and manipulate partial and uncertain information. Starting from the work of Tritchler and Lockwood [8], the paper proposes a general model of mass allocation and combination, and shows how to practically derive a complete or a partial ranking of the alternatives. A small example is provided as an illustration.

Introduction

The aim of the paper is to study the task of constructing a linear order, or a ranking, of n alternatives, based on paired comparisons. Paired experiments consist in presenting two objects to one or several judges and asking them to choose the best alternative among the pair. Each paired comparison is supposed to provide uncertain pieces of evidence on the ranking relation, and the derivation of a linear order is considered as an information fusion problem. Uncertain possibilistic preferences have been already considered e.g. in [START_REF] Dubois | On the ranking of ill-known values in possibility theory[END_REF]. In this paper, the problem is tackled in the framework of belief functions.

A first work using belief functions to describe the uncertainty about the comparisons has been proposed in [START_REF] Tritchler | Modelling the reliability of paired comparisons[END_REF]. Unfortunately, this work remains essentially theoretical and gives very few tools for practical applications. Our paper synthesizes their main results in Section 3 and extends them in Section 4 in three ways: a more general model of mass allocation is proposed, a linear programming approach for determining the most plausible ranking is introduced, and a heuristic procedure for choosing only a partial order, starting from the most plausible ranking, is given. A small example in Section 5 illustrates the proposed method. Note that, due to space limitations, basic knowledge on belief functions will be assumed. The reader is referred, in particular, to [START_REF] Shafer | A mathematical theory of evidence[END_REF][START_REF] Smets | The Transferable Belief Model[END_REF].

Basic notions on relations

Let O = {o 1 , o 2 , ..., o n } be a set of n alternatives. We recall that a strict order R on O is a binary relation for which the following properties hold for all o i , o j and o k ∈ O:

• if (o i , o j ) ∈ R then (o j , o i ) ∈ R (asymmetry); • if (o i , o j ) ∈ R and (o j , o k ) ∈ R then (o i , o k ) ∈ R (transitivity); If the order is complete (either (o i , o j ) ∈ R or (o j , o i ) ∈ R), it is a linear (or total) order, otherwise it is a partial order. If (o i , o j ) ∈ R or (o j , o i ) ∈ R then o i
and o j are comparable, otherwise they are said incomparable.

A linear order L is called a linear extension of a partial order

P if P ⊆ L (∀ (o i , o j ) ∈ P, then (o i , o j ) ∈ L).
To each partial order can thus be associated the set of its linear extensions. Conversely, any collection C of total orders defines a partial order H as follows:

(o i , o j ) ∈ H iff (o i , o j )

Pairwise comparisons in the framework of belief functions

Combining pairwise comparisons in the framework of belief functions has been already addressed by Tritchler and Lockwood [8]. This section follows their presentation and synthetizes the most useful notions. They consider that, for each pair (o i , o j ) of alternatives in O (1 ≤ i < j ≤ n), an expert expresses its preference between o i and o j using a mass function m Θ i j quantifying the uncertainty in the evaluation. This mass function is defined on the frame of discernement Θ i j = {o i ≻ o j , o j ≻ o i }: the singleton o i ≻ o j means that o i should be ranked before o j and the singleton o j ≻ o i that o j should be ranked first. Tritchler and Lockwood propose to use a simple support mass function: the expert chooses one of the singletons with mass α i j and the rest of the mass is allocated to Θ i j . The value α i j is interpreted as the reliability of the choice.

Let φ i j denote a focal element of m Θ i j . Each focal element φ i j has a graph repre- sentation which consists of two nodes, o i and o j , with one arc if φ i j is a singleton element, and no arc if the focal element is Θ i j .

The problem is to derive from the n(n -1)/2 mass functions a ranking of the alternatives. This task may be seen as an information fusion problem and Dempster's rule of combination [START_REF] Shafer | A mathematical theory of evidence[END_REF] can be used to this end. Let I denote the set {(i, j) | 1 ≤ i < j ≤ n} and let Θ (I) denote the product space:

Θ (I) = Θ 12 ×Θ 13 × ... ×Θ (n-1)n
Θ (I) consists of all complete asymmetric relations (or graphs) defined on the set O.

Before being combined, pieces of evidence from all pairs have to be expressed on the same frame of discernement, namely the product space Θ (I). This is achieved by applying the vacuous extension operation [START_REF] Shafer | A mathematical theory of evidence[END_REF][START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized Bayesian theorem[END_REF] to each m Θ i j . This operation, denoted ↑, transfers each mass m Θ i j (φ i j ) to φ i j × Θ (I -{(i, j)}). The symbol ⊕ representing Dempster's rule of combination, the expression of the combination can thus be formally written as:

m Θ (I) = m Θ 12 ↑Θ (I) ⊕ m Θ 13 ↑Θ (I) ⊕ ... ⊕ m Θ (n-1)n ↑Θ (I) , (1)
or, using the commonalities:

q Θ (I) = ∏ (i, j)∈I q Θ i j ↑Θ (I) . (2)
The focal elements of m Θ (I) are of the form: φ = φ 12 × φ 13 × ... × φ (n-1)n , where φ i j is a focal element of m Θ i j and the mass resulting from the combination is:

m Θ (I) (φ ) = m Θ 12 (φ 12 )m Θ 13 (φ 13 )...m Θ (n-1)n (φ (n-1)n ). (3)
In terms of graph, each focal element φ of m Θ (I) can be represented by a directed graph formed by the union of individual graphs. Since each φ i j is equal either to a singleton or to Θ i j , each focal element φ is a subset composed of complete asymmetric relations on O, whose graphs contain the arcs of φ . The combination described above allocates masses on various sets of asymmetric relations defined on O. A first objective is to find a linear ordering on O that is the most compatible with the pairwise evaluations. This can be done by imposing conditions on the set in which the solution has to be found. Let L denote the set of all linear orders defined on O which is a subset of Θ (I). To impose the nature of the solution, it is proposed in [START_REF] Tritchler | Modelling the reliability of paired comparisons[END_REF] to condition the mass m Θ (I) with respect to L :

m Θ (I) [L ] = m Θ (I) ⊕ m L , (4)
with m L a categorical mass function defined by m L (L ) = 1. Expressed using the commonalities, the whole combination can be written as:

q Θ (I) [L ] = 1 1 -K q L ∏ (i, j)∈I q Θ i j ↑Θ (I) , ( 5
)
where K is the conflict resulting from the combination of m Θ (I) with m L . K can be interpreted as an index of the internal coherence of the evaluations. Its practical computation will be explained when dealing with partial orders.

Practical use

We consider in this section a general form of mass allocation defined by:

   m Θ i j (o i ≻ o j ) = α i j , m Θ i j (o j ≻ o i ) = β i j , m Θ i j (Θ i j ) = 1 -α i j -β i j . (6)
This mass allocation may come from a single expert who is asked to provide, for each (i, j) ∈ I, the above mass function, or from the combination of the evaluations of several experts. In that case, for each (i, j) ∈ I, several m Θ i j k are available and they have to be fused to provide m Θ i j . The choice of the combination rule depends on the hypotheses made on the dependence between the experts. If they can be considered as independent, Dempster's rule should be chosen. Otherwise, the cautious rule [START_REF] Denoeux | Conjunctive and Disjunctive Combination of Belief Functions Induced by Non Distinct Bodies of Evidence[END_REF] may be preferred.

Most plausible ranking

Let L ∈ L be a strict linear ordering on O. L being a singleton of the frame of discernement, one has q L ({L}) = 1 and, with the mass allocation ( 6), one has:

q Θ i j ↑Θ (I) ({L}) = 1 -β i j if (o i , o j ) ∈ L, q Θ i j ↑Θ (I) ({L}) = 1 -α i j if (o j , o i ) ∈ L. ( 7
)
Let us introduce n(n -1)/2 binary variables l i j ((i, j) ∈ I) defined by l i j = 1 if (o i , o j ) ∈ L and 0 otherwise. Using ( 7) and ( 5), the commonality, or, equivalently, the plausiblity of L can be written as:

q Θ (I) [L ]({L}) = pl Θ (I) [L ]({L}) = 1 1 -K ∏ (i, j)∈I (1 -β i j ) l i j (1 -α i j ) 1-l i j , ( 8
)
where K is the conflict resulting from the combination of m Θ (I) with m L . To find the most plausible ranking of the alternatives, it is not necessary to enumerate all possible linear orderings. We propose to solve the problem using a linear programming approach. Maximizing expression ( 8) is equivalent to maximize its logarithm so that, omitting the constant term depending on K, the most plausible ranking L can be found as the solution of the following linear program:

max l i j ∈{0,1} ∑ (i, j)∈I l i j ln 1 -β i j 1 -α i j , (9)
subject to:

l i j + l jk -1 ≤ l ik , ∀i < j < k, l ik ≤ l i j + l jk , ∀i < j < k. ( 10
)
The constraints are used to insure that L belongs to L : if l i j = 1 and l jk = 1 then l ik = 1. If l i j = 0 and l jk = 0 then l ik = 0.

Remark 1 Note that the general form of mass allocation (6) allows us to take naturally into account tied evaluations. If the comparison between o i and o j results in a tie

, we let α i j = β i j . Then, it can be easily seen that the pair (o j , o j ) does not appear any more in the objective function.

Plausibility of a partial ranking

In some situations, it may be also interesting to compute the plausiblity of a partial order. When working in the set of asymmetric relations as defined by Tritchler and Lockwood, it not possible to provide an analytical expression. However, some results from [START_REF] Tritchler | Modelling the reliability of paired comparisons[END_REF] make it possible to use simple algorithms based on graph theory to compute the plausibility of any partial order. The following theorem is proved in [START_REF] Tritchler | Modelling the reliability of paired comparisons[END_REF]:

Theorem 1 (Tritchler and Lockwood (1991)) Let K be the conflict between the two mass functions m Θ (I) and m L .

1. K = ∑ m Θ (I) (φ ) where the summation is over every focal element of m Θ (I) whose graph contains a cycle; 2. Let H be a partial order realized by a focal element θ H of m Θ (I) [L ]. Then θ H is the set of all linear extensions of H (θ H is the largest subset of L which realizes H).

m Θ

(I) [L ](θ H ) = 1 1-K ∑ m Θ (I) (φ )
, where the summation is over every focal element φ of m Θ (I) such that the transitive closure of G(φ ), G t (φ ), is equal to H.

To compute the plausiblity of any partial order, one has to sum the masses associated to all focal elements with a non null intersection with this partial order. The following lemma helps to recognize the focal elements which intersect a given partial order: Computing the plausibility of any partial order H is thus achieved by summing the masses of all focal sets θ H ′ such that H ′ ∪ H is acyclic. . Lemma 1 and Theorem 1 allow us to propose two simple procedures (one exact, one approximate) for computing the plausibility of a given partial order. The two procedures are detailed in Algorithms 1 and 2.

Heuristic search for a partial ranking

If the plausibility of the most plausible ranking is too low, it can be preferable to provide the user with only a partial ranking of the alternatives. The algorithms de-Algorithm 1 Plausibility of a partial order H 1: K ← 0 2: pl(H) ← 0 3: for all φ = φ 12 × φ 13 × ... × φ (n-1)n do 4:

Compute the transitive closure G t (φ ) of G(φ ) 5:

Compute the mass m = m Θ (I) (φ ) by equation (3) 6:

if G t (φ ) contains a cycle then 7:

K = K + m 8: else if G(H) ∪ G t (φ ) is acyclic then pl(H) = pl(H) + m 9: end if 10: end for 11: pl(H) ← 1 1-K pl(H)
Algorithm 2 Approximate computation by a Monte-Carlo approach

1: N pl ← 0 2: N K ← 0 3: for rep ← 1, N do 4: G ← / 0 5:
for each (i, j) ∈ I do 6:

With probability m Θ i j (φ i j ), randomly select a focal element φ i j from the focal ele- ments of m Θ i j 7:

if φ i j is a singleton, add the corresponding arc to G 8:

end for 9:

Compute the transitive closure G t of G 10:

if G t contains a cycle then 11:

N K = N K + 1 12: else if G(H) ∪ G t is acyclic then N pl = N pl + 1 13: end if 14: end for 15: K = N K N 16: pl(H) = 1 1 - K N pl N
scribed in the previous section allow us to compute the plausibility of any partial order. Instead of exploring every possible partial orders, which would be practically intractable, we propose a heuristic procedure based on a principle of hierarchical clustering. We start from the most plausible ranking (see Section 4.1). Then, at the first step of the procedure, we compute the plausibility of every partial orders obtained by removing one pair of adjacent alternatives from the total order relation. The most plausible partial order is retained and the corresponding pair of alternatives is "merged". The process is repeated until all alternatives have been merged into a single one. It is easy to see that the sequence of plausibility values is monotonically increasing. Finally, a partial order with the desired level of plausibility can be chosen by the user.

Example

We illustrate the methods described above using an example inspired from [START_REF] Tritchler | Modelling the reliability of paired comparisons[END_REF]. In a study conducted at the Ontario Cancer Institute, subjects were asked to give their preferences about four scenarios describing ethical dilemmas in health care. The preferences for all six possible scenario pairs were obtained. The experts were also asked to rate the reliability of their evaluations. The preferences of a subject can be represented by a directed graph in which the vertices are the scenarios and the edges represent the relation "is preferred to". The values on the edges represent the reliability given by the expert. The graphs of the experts are given in Figure 1. The fact that graph 1 (left) contains a cycle (ADB) shows that the evaluations of expert 1 are not fully consistent. There is no cycle in graph 2 (right), but the degrees of belief are weaker than for expert 1. The evaluations of each expert are modelled using the mass allocation expressed by equation ( 6) with either α i j or β i j equal to zero. Applied individually to each expert, the procedure for deriving a complete ranking of the alternatives (Section 4.1) gives the ranking A ≻ D ≻ B ≻ C with a plausiblity of 0.8070 for the first expert, and the ranking A ≻ C ≻ D ≻ B with a plausibility equal to 1 for the second one. The plausiblities thus reflect the internal coherence of the experts. The evaluations of the experts can also be combined before searching for a complete ranking. The results of the combination using Dempster's rule of combination can be found in Table 1. The most plausible total ranking derived from Table 1 is A ≻ D ≻ B ≻ C with a plausibility equal to 0.8893. Applying the heuristic procedure using the masses of Table 1 for determining a partial ranking gives the result presented in Figure 2. The dendrogram can be cut at the desired level of plausibility. For example, the partial order A ≻ D ≻ {B,C} reaches a plausibility of almost 0.96. Fig. 2 Dendrogram obtained from Table 1.

Conclusion

In this paper, we have has shown how to use the framework of belief functions to model paired comparisons and how to derive from these individual judgements a total or a partial ranking of the alternatives. The linear order is obtained by solving a linear program maximizing the plausibility of the relation. A heuristic procedure has been proposed to provide only a partial order when the plausibility of the linear order is too low. This work offers several perspectives, among which the application of the approach to machine learning problems like instance or label ranking problems.

Lemma 1 (

 1 Tritchler and Lockwood (1991)) Let C 1 and C 2 be two subsets of L realizing the partial orders H 1 and H 2 . Then C 1 ∩ C 2 = / 0 if and only if H 1 ∪ H 2 is acyclic.

Fig. 1

 1 Fig. 1 Graph representation of the evaluations; (left) : expert 1 ; (right) : expert 2.

  belongs to all linear order in C. One then says that H is realized by C. Note that two subsets can realize the same partial order.Any relation R can be conveniently represented by a directed graph with nodes O. Two nodes o i and o j are connected by an arc in the graph if (o i , o j ) ∈ R.

Table 1

 1 Mass assignment using Dempster's rule of combination(o i , o j ) o i ≻ o j o j ≻ o i Θ i j

	(A,B) 0.3056 0.3056 0.3889
	(A,C) 0.9991	0	0.0009
	(A,D) 0.9964	0	0.0036
	(B,C) 0.7266 0.2187 0.0547
	(B,D)	0	0.9324 0.0676
	(C,D) 0.0594 0.0094 0.9312