Free quantitative fourth moment theorems on Wigner space

0.23 MB·

by  Solesne Bourguin

FREE QUANTITATIVE FOURTH MOMENT THEOREMS ON
WIGNER SPACE
7
1
SOLESNEBOURGUINANDSIMONCAMPESE
0
2

n
Abstract. WeproveaquantitativeFourthMomentTheoremforWignerin-
a
tegralsofanyorderwithsymmetrickernels,generalizinganearlierresultfrom
J
Kemp et al. (2012). The proof relies on free stochastic analysis and uses
9
a new biproduct formula for bi-integrals. A consequence of our main result
1
is a Nualart-Ortiz-Latorre type characterization of convergence in law to the

semicircular distribution for Wigner integrals. As an application, we provide
]
Berry-EsseentypeboundsinthecontextofthefreeBreuer-Majortheoremfor
A
thefreefractionalBrownianmotion.
O
.
h
t
a
m
1. Introduction
[

1 Let (A,ϕ) be a tracial W∗-probability space, S be a semicircular random vari-
v
able and F = I (f) be a self-adjoint Wigner integral (for a simple example, take
4 n
off-diagonal homogeneous sums of a semicircular system). Recently, Kemp et al.
1
4 showed in [KNPS12] that for a sequence of such Wigner integrals, convergence of
5 the fourth moment controls convergence in distribution towards the semicircular
0 law. Moreover, they provided a quantitative bound in terms of the free gradient
.
1 operator, which is of the form (all unexplained notation appearing in this section
0 will be introduced in the sequel)
7
1 1
(1) d (F,S) ϕ ϕ N−1F ♯( F)∗ds 1 1 .
v: C2 ≤ 2 ⊗ ∇s 0 ∇s − ⊗
(cid:18)(cid:12)Z (cid:12)(cid:19)
i (cid:12) (cid:0) (cid:1) (cid:12)
X Here,d isadistancethatmetriz(cid:12)esfreeconvergenceindistribution(cid:12)(seeDefinition
C2 (cid:12) (cid:12)
r 2.4), denotesthefreegradientoperatorfirstintroducedbyBianeandSpeicherin
a [BS98∇] and N−1 stands for the pseudo-inverse of the number operator (see Section
0
2). In the special case of Wigner integrals of order two, Kemp et al. showed
in [KNPS12] that the gradientexpressionappearing in (1) can further be bounded
by the fourth moment. To be more precise, it holds that
1
d (I (f),S) ϕ ϕ N−1I (f) ♯( I (f))∗ds 1 1
C2 2 ≤ 2 ⊗ ∇s 0 2 ∇s 2 − ⊗
(2) (cid:18)(cid:12)Z (cid:12)(cid:19)
(cid:12) (cid:0) (cid:1) (cid:12)
1 3 (cid:12) (cid:12)
ϕ(cid:12)(I (f)4) 2. (cid:12)
2
≤ 2 2 −
r
p
2010 Mathematics Subject Classification. 46L54,68H07,60H30.
Key words and phrases. Free probability, Wigner integrals, free Malliavin calculus, free sto-
chasticanalysis,freequantitative centrallimittheorems,freeFourthMomentTheorems.
1
2 SOLESNEBOURGUINANDSIMONCAMPESE
A question left open in the aforementioned article is whether a similar fourth mo-
mentboundholdsforWignerintegralsofhigherorders,asisthecaseinthecommu-
tative setting (see Nualart and Peccati [NP05] and Nourdin and Peccati[NP09b]).
In this paper, we provide a positive answer to this question by proving fourth mo-
ment bounds for Wigner integrals of any order with symmetric kernels. Our main
result can be paraphrased as follows (see Theorem 3.7 for a precise statement).
Theorem. For a Wigner integral F of order n with normalized symmetric kernel
it holds that
2
ϕ ϕ N−1F ♯( F)∗ds 1 1 C ϕ F4 2 .
⊗ (cid:12) R ∇s 0 ∇s − ⊗ (cid:12) ≤ n −
(cid:12)(cid:12)Z + (cid:0) (cid:1) (cid:12)(cid:12) (cid:16) (cid:0) (cid:1) (cid:17)
(cid:12) (cid:12) 
The constant C(cid:12) grows asymptotically linearly with(cid:12) n and is a local maximum of
n
a certain polynomial (see Theorem 3.7 for full details). Combined with (1), our
result quantifies the free Fourth Moment Theorem [KNPS12, Theorem 1.3] for the
case of Wigner integrals with symmetric kernels. In particular, √C = 1 3, so
2 2 2
that, by Cauchy-Schwarz,the bound (2) is included as a special case. q
It is well-known that in order to ensure that Wigner integrals are self-adjoint (and
thus free random variables), the symmetry of the kernel can be relaxed to mirror-
symmetry (see Definition 2.7). As our main bound is stated for symmetric kernels,
thenaturalquestionariseswhetherornotitcanbegeneralizedtocoverthemirror-
symmetriccase aswell. The answerto this questionis negative,as is shownby the
counterexample in Remark 3.9.
In the proof of our main result we use a new biproduct formula (see Theorem
3.5) for Wigner bi-integrals (see Subsection 2.2) which generalizes the product for-
mula proved by Biane and Speicher in [BS98] for usual Wigner integrals. In this
biproductformula,thenestedcontractionsbecomewhatwecallbicontractions. As
product formulae play a central role in free (and also classical) stochastic analysis,
this might be of independent interest. Other ingredients include the free Malliavin
calculus introduced by Biane and Speicher in [BS98] as well as a fine combinato-
rial analysis. A direct consequence of our bound is a Nualart-Ortiz-Latorre type
equivalent condition for convergence towards the semicircular law, which reads as
follows (see Theorem 3.10 for a precise statement).
Theorem. AsequenceF ofWignerintegralsofordernwithnormalizedsymmetric
k
kernels converges in law to the standard semicircular distribution if, and only if,
( F )♯( F )∗ds n 1 1 in L2(A A,ϕ ϕ).
s k s k
R ∇ ∇ → · ⊗ ⊗ ⊗
Z +
This is a free analogue of the main result of [NOL08].
Our findings contribute to the growing literature on free limit theorems obtained
by means of free Malliavin calculus and free stochastic analysis. Earlier results
include the already mentioned free Fourth Moment Theorem for multiple Wigner
integrals [KNPS12], its multidimensional extension [NPS13], the free Fourth Mo-
ment Theorem for free Poisson multiple integrals proved in [BP14b] and [Bou16],
FREE QUANTITATIVE FOURTH MOMENT THEOREMS 3
free non-central limit theorems for Wigner and free Poisson integrals obtained
in [DN12], [NP13] and [Bou15], as well as limit theorems for the q-Brownian mo-
tion [DNN13] and convergenceof free processes[NT14]. However,all these results,
with the exception of [KNPS12] for the case of second order Wigner integrals,
are not quantitative. In the commutative setting, which inspired this line of re-
search in the context of free probability theory, the picture is much more com-
plete. Here, quantitative limit theorems existin the frameworkof Wiener integrals
([NP05, PT05, NOL08, NP09b, NPR10, NP09a] and references therein), Poisson
integrals([PSTU10, PZ10,Pec11, BP14a,PT13]andreferencestherein)andeigen-
functions of diffusive Markov generators ([Led12, ACP14, CNPP16]).
Therestofthispaperisorganizedasfollows: Section2introducesthebasicconcepts
of free probability theory and free stochastic analysis. The biproduct formula,
our fourth moment bound, as well as the Nualart-Ortiz-Latorre characterization
are presented and proved in Section 3. We conclude by providing a Berry-Esseen
bound for the free Breuer-Major theorem for the free fractional Brownian motion
in Section 4.
2. Preliminaries
2.1. Elements of free probability. Inthe following,ashortintroductionto free
probabilitytheoryisprovided. Forathoroughandcompletetreatment,see[NS06],
[VDN92] and [HP00]. Let (A,ϕ) be a tracial W∗-probability space, that is A is a
vonNeumannalgebrawithinvolution andϕ: A Cisaunitallinearfunctional
∗ →
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(YX) for all X⇒,Y A).
The self-adjo∈int elements of A will be referred to as random variables. Gi∈ven a
random variable X A, the law of X is defined to be the unique Borel measure
on R having the sam∈e moments as X (see [NS06, Proposition 3.13]). The non-
commutative space L2(A,ϕ) denotes the completion of A with respect to the
norm X = ϕ(XX∗).
k k2
Definition 2.1p. A collection of random variables X ,...,X on (A,ϕ) is said to
1 n
be free if
ϕ([P (X ) ϕ(P (X ))] [P (X ) ϕ(P (X ))])=0
1 i1 − 1 i1 ··· m im − m im
whenever P ,...,P are polynomials and i ,...,i 1,...,n are indices with
1 m 1 m
∈ { }
no two adjacent i equal.
j
Let X A. The k-th moment of X is given by the quantity ϕ(Xk), k N .
0
Nowass∈ume thatX is aself-adjointbounded elementofA (inother words,X∈ is a
bounded randomvariable), andwrite ρ(X)= X [0, )to indicate the spectral
k k∈ ∞
radius of X.
Definition 2.2. The law (or spectral measure) of X is defined as the unique Borel
probability measure µ on the real line such that P(t) dµ (t) = ϕ(P(X)) for
X R X
everypolynomialP R[X]. Aconsequenceofthisdefinitionisthatµ hassupport
X
∈ R
in [ ρ(X),ρ(X)].
−
4 SOLESNEBOURGUINANDSIMONCAMPESE
The existence and uniqueness of µ in such a general framework are proved e.g.
X
in [Tao12, Theorem 2.5.8] (see also [NS06, Proposition 3.13]). Note that, since µ
X
has compact support, the measure µ is completely determined by the sequence
X
ϕ(Xk): k 1 .
≥
L(cid:8)et X : n 1(cid:9) beasequenceofnon–commutativerandomvariables,eachpossibly
n
belo{nging to≥a d}ifferent non-commutative probability space (A ,ϕ ).
n n
Definition 2.3. The sequence X : n 1 is said to converge in distribution
n
to a limiting non-commutative r{andom≥var}iable X (defined on (A ,ϕ )), if
∞ ∞ ∞
lim ϕ (P(X ))=ϕ (P(X )) for every polynomial P R[X].
n→+∞ n n ∞ ∞
∈
If X ,X are bounded (and therefore the spectral measures µ ,µ are well-
n ∞ Xn X∞
defined), this last relation is equivalent to saying that
P(t)µ (dt) P(t)µ (dt).
R Xn → R X∞
Z Z
Anapplicationofthemethodofmomentsyieldsimmediatelythat,inthiscase,one
has also that µ weakly converges to µ , that is µ (f) µ (f), for every
f :R R bounXdned and continuous (noteXt∞hat no addiXtinonal u→nifoXrm∞ boundedness
→
assumption is needed).
Let h(x) = eixξν(dξ) be the Fourier transform of a complex measure ν on R.
R
Note that, as ν is finite, h is continuous and bounded. For such functions h, define
the seminormR I (h) by
2
I (h)= ξ2 ν (dξ).
2
R | |
Z
Let denotethesetofthosefunctionshforwhichI (h)< . Usingtheseminorm
2 2
I aCnd the set of functions , one can define a distance b∞etween two self-adjoint
2 2
C
random variables.
Definition2.4. Fortwoself-adjointrandomvariablesX,Y,thedistanced (X,Y)
C2
between X and Y is defined as
d (X,Y)=sup ϕ(h(X)) ϕ(h(Y)) : h , I (h) 1 .
C2 {| − | ∈C2 2 ≤ }
Asisprovedin[KNPS12],thedistanced isweakerthantheWassersteindistance
C2
but still metrizes convergence in law.
Definition 2.5. The centered semicircular distribution with variance t > 0, de-
noted by (0,t), is the probability distribution given by
S
(0,t)(dx)=(2πt)−1 4t x2dx, x <2√t.
S − | |
Definition 2.6. A free BrownianmotiopnS consists of: (i) a filtration A : t 0
t
of von Neumann sub-algebras of A (in particular, A A for 0 s{< t), (≥ii) a}
s t
collectionS = S : t 0 of self-adjointoperators in A⊂such that:≤(a) S =0 and
t 0
S A for all{t 0,≥(b)}for all t 0, S has a semicirculardistribution with mean
t t t
∈ ≥ ≥
zero and variance t, and (c) for all 0 u < t, the increment S S is free with
t u
respect to A , and has a semicircula≤r distribution with mean z−ero and variance
u
t u.
−
FREE QUANTITATIVE FOURTH MOMENT THEOREMS 5
For every integer n 1, the space L2 Rn;C =L2 Rn denotes the collection of
all complex-valued f≥unctions on Rn that+are square-int+egrable with respect to the
Lebesgue measure on Rn. + (cid:0) (cid:1) (cid:0) (cid:1)
+
Definition 2.7. Let n be a natural number and let f be a function in L2 Rn .
+
(1) The adjoint of f is the function f∗(t ,...,t )=f(t ,...,t ). (cid:0) (cid:1)
1 n n 1
(2) The function f is called mirror-symmetric if f =f∗, i.e., if
f(t ,...,t )=f(t ,...,t )
1 n n 1
for almost all (t ,...,t ) Rn with respect to the product Lebesgue mea-
1 n ∈ +
sure.
(3) The function f is called (fully) symmetric if it is real-valued and, for any
permutation σ in the symmetric group S , it holds that f(t ,...,t ) =
n 1 n
f t ,...,t foralmostall(t ,...,t ) Rn withrespecttotheprod-
σ(1) σ(n) 1 n ∈ +
uct Lebesgue measure.
(cid:0) (cid:1)
Definition 2.8. Let n,m be natural numbers and let f L2 Rn and g
∈ + ∈
L2 Rm . Letp n mbeanaturalnumber. Thep-thnestedcontractionf ⌢p g of
+ ≤ ∧ (cid:0) (cid:1)
f a(cid:0)nd g(cid:1)is the L2 Rn++m−2p function defined by nested integration of the middle
p variables in f (cid:16)g: (cid:17)
⊗
p
f ⌢g(t ,...,t ) = f(t ,...,t ,s ,...,s )
1 n+m−2p 1 n−p 1 p
Rp
Z +
g(s ,...,s ,t ,...,t )ds ds .
p 1 n−p+1 n+m−2p 1 p
···
0
In the case where p=0, the function f ⌢g is just given by f g.
⊗
Forf L2 Rn ,wedenotebyI (f)themultipleWignerintegraloff withrespect
∈ + n
tothefreeBrownianmotionasintroducedin[BS98]. ThespaceL2( ,ϕ)= I (f):
f L2(Rn)(cid:0),n (cid:1) 0 isaunital -algebra,withproductrulegiven,foSranyn,{mn 1,
∈ + ≥ } ∗ ≥
f L2 Rn , g L2 Rm , by
∈ + ∈ +
(cid:0) (cid:1) (cid:0) (cid:1) n∧m
p
(3) I (f)I (g)= I f ⌢g
n m n+m−2p
Xp=0 (cid:16) (cid:17)
and involution I (f)∗ = I (f∗). For a proof of this formula, see [BS98]. Fur-
n n
thermore, as is well-known, multiple integrals of different orders are orthogonal in
L2(A,ϕ), whereas for two integrals of the same order, the Wigner isometry
(4) ϕ(I (f)I (g)∗)= f,g .
n n h iL2(Rn+)
holds.
Remark 2.9. Observe that it follows from the definition of the involution on the
algebraL2( ,ϕ) that operatorsof the type I (f) are self-adjointif and only if f is
n
S
mirror-symmetric.
6 SOLESNEBOURGUINANDSIMONCAMPESE
2.2. Bi-integrals and free gradient operator. This subsection introduces the
notionofbi-integralandtheactionofthefreegradientoperatoronWignerintegrals.
For a full treatment of these objects, see [BS98].
Let n,m be two positive integers and f = g h L2 Rn L2 Rm . Then, the
⊗ ∈ + ⊗ +
Wigner bi-integral I I (f) is defined as
n m
⊗ (cid:0) (cid:1) (cid:0) (cid:1)
I I (f)=I (g) I (h).
n m n m
⊗ ⊗
This definition is extended linearly to generic elements f L2 Rn L2 Rm =
∈ + ⊗ + ∼
L2 Rn+m . From the Wigner isometry (4) for multiple integrals,we obtain the so
+ (cid:0) (cid:1) (cid:0) (cid:1)
call(cid:0)edWig(cid:1)nerbisometry: forf ∈L2 Rn+ ⊗L2 Rm+ andg ∈L2 Rn+′ ⊗L2 Rm+′
it holds that (cid:0) (cid:1) (cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:17)
(5)
f,g if n=n′ and m=m′,
ϕ ϕ(In Im(f)In′ Im′(g)∗)= h iL2(Rn+)⊗L2(Rm+)
⊗ ⊗ ⊗ (0 otherwise
Remark 2.10. Observethat,foranynaturalnumbersn,mandanyfunctiong h
L2 Rn L2 Rm , it holds that ⊗ ∈
+ ⊗ +
(cid:0) (cid:1) In(cid:0) Im(cid:1)(g h)∗ =(In(g) Im(h))∗ =In(g)∗ Im(h)∗
⊗ ⊗ ⊗ ⊗
=I (g∗) I (h∗)=I I (g h)∗ ,
n m n m
⊗ ⊗ ⊗
sothatthe operatorIn Im(g h)isself-adjointifandonl(cid:0)yifboth(cid:1)thefunctiong
⊗ ⊗
andharemirror-symmetric. Bycontinuousextension(usingtheWignerbisometry
(5)), it holds that for any fully symmetric function f L2 Rn L2 Rm , the
∈ + ⊗ +
operator I I (f) is self-adjoint.
n m
⊗ (cid:0) (cid:1) (cid:0) (cid:1)
Let(A,ϕ)beaW∗-probabilityspace. AnA A-valuedstochasticprocesst U
t
iscalledabiprocess. Forp 1, U isaneleme⊗ntofB ,the spaceofLp-biproce7→sses,
p
≥
if its norm
∞
U 2 = U 2 dt
k kBp k tkLp(A⊗A,ϕ⊗ϕ)
Z0
is finite.
The free gradient operator : L2( ,ϕ) B is a densely-defined and closable
2
∇ S →
operator whose action on Wigner integrals is given by
n
I (f)= I I f(k) ,
∇t n k−1⊗ n−k t
Xk=1 (cid:16) (cid:17)
where f(k)(x ,...,x )=f(x ,...,x ,t,x ,...,x ) is viewed as an element
t 1 n−1 1 k−1 k n−1
of L2 Rk−1 L2 Rn−k .
+ ⊗ +
Rema(cid:0)rk 2.1(cid:1)1. For(cid:0)genera(cid:1)l elements of L2( ,ϕ) in its domain, the free gradient
S
is customarily defined via a Fock space construction (see [BS98]). This level of
generality will not be needed in the sequel.
We will also make use of the pseudo-inverse of the number operator N−1, whose
0
action on a multiple Wigner integral of order n 1 is given by N−1I (f) =
≥ 0 n
1I (f).
n n
FREE QUANTITATIVE FOURTH MOMENT THEOREMS 7
Before concluding this section, we introduce ♯ to be the associative action of A
Aop (where Aop denotes the opposite algebra) on A A, as ⊗
⊗
(6) (A B)♯(C D)=(AC) (DB).
⊗ ⊗ ⊗
Furthermore, we also write ♯ to denote the action of A L2(R ) Aop on A
+
L2(R ) A, as ⊗ ⊗ ⊗
+
⊗
(A f B)♯(C g D)=(AC) fg (DB).
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
The multiplication ♯ naturally appears in the following bound from [KNPS12] on
the d distance introduced above.
C2
Theorem2.12([KNPS12]). LetS beastandardsemicircular random variable and
F L2( ,ϕ) be self-adjoint, in the domain of the free gradient and such that
∈ S ∇
ϕ(F)=0. Then,
1
(7) d (F,S) ϕ ϕ N−1F ♯( F)∗ds 1 1 .
C2 ≤ 2 ⊗ (cid:12) R ∇s 0 ∇s − ⊗ (cid:12)!
(cid:12)Z + (cid:12)
(cid:12) (cid:0) (cid:1) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3. Main results
3.1. Bicontractions and biproduct formula. As announced in the introduc-
tion, we will need an extension of the product formula (3) from [BS98]. To this
end, we introduce the notion of bicontraction.
Definition 3.1. Let n ,m ,n ,m be positive integers. Let f L2 Rn1
1 1 2 2 ∈ + ⊗
L2 Rm1 = L2 Rn1+m1 and g L2 Rn2 L2 Rm2 = L2 Rn2+m2 and let
+ ∼ + ∈ + ⊗ + ∼ + (cid:0) (cid:1)
p,r
p n n , r m m be natural numbers. The (p,r)-bicontraction f ⌢ g
≤(cid:0) 1 ∧(cid:1) 2 ≤(cid:0) 1 ∧ (cid:1)2 (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
is the L2 Rn1+n2−2p L2 Rm1+m2−2r = L2 Rn1+n2+m1+m2−2p−2r function
+ ⊗ + ∼ +
defined by(cid:16) (cid:17) (cid:0) (cid:1) (cid:16) (cid:17)
p,r
f ⌢g(t ,...,t )
1 n1+n2+m1+m2−2p−2r
= f(t ,...,t ,s ,...,s ,y ,...,y ,
1 n1−p p 1 1 r
Rp+r
Z +
t ,...,t )
n1+n2+m2−2p−r+1 n1+n2+m1+m2−2p−2r
g(s ,...,s ,t ,...,t ,y ,...,y )
× 1 p n1−p+1 n1+n2+m2−2p−r r 1
ds ds dy dy .
1 p 1 r
··· ···
Remark 3.2. Observe that for f = f f and g = g g with f L2 Rn1 ,
1 ⊗ 2 1 ⊗ 2 1 ∈ +
f L2 Rm1 , g L2 Rn2 and g L2 Rm2 , the above definition reads
2 ∈ + 1 ∈ + 2 ∈ + (cid:0) (cid:1)
(8) (cid:0) f(cid:1)p⌢,r g =(f(cid:0) f(cid:1))p⌢,r (g g )(cid:0)= f(cid:1) ⌢p g g ⌢r f ,
1 2 1 2 1 1 2 2
⊗ ⊗ ⊗
(cid:16) (cid:17) (cid:16) (cid:17)
wherethecontractionsappearingontheright-handsidearethenestedcontractions
introduced in Definition 2.8.
p,r
Remark 3.3. In what follows, for f,g as in Definition 3.1, we write f ⌢ g and
s
f ⌢ g to denote the bicontraction and contraction of f and g, respectively. Here,
we have somewhat abused notation by using the same symbol for a function living
inL2 Rn1 L2 Rm1 oritsidentificationinL2 Rn1+m1 . However,itwillalways
+ ⊗ + +
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
8 SOLESNEBOURGUINANDSIMONCAMPESE
be clear from the type of contraction used which version of the function is being
considered.
The following result collects some properties of bicontractions in the case where
both functions are symmetric.
Lemma 3.4. For n ,m ,n ,m N, let f L2 Rn1 L2 Rm1 =L2 Rn1+m1
1 1 2 2 ∈ ∈ + ⊗ + ∼ +
andg L2 Rn2 L2 Rm2 =L2 Rn2+m2 befullysymmetricfunctions. Further-
∈ + ⊗ + ∼ + (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
more, let p n n and r m m be natural numbers such that p+r =p′+r′.
1 2 1 2
(cid:0)≤ (cid:1)∧ (cid:0) (cid:1)≤ ∧(cid:0) (cid:1)
Then, the following is true.
p,r p+r
(i) f ⌢g =f ⌢ g.
∼
p,r p′,r′
(ii) f ⌢g =f ⌢ g.
p,r 2 p+r 2
(iii) f ⌢g = f ⌢ g .
L2(Rn1+n2−2p)⊗L2(Rm1+m2−2r) L2(Rn1+n2+m1+m2−2p−2r)
+ + +
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
(iv) f(cid:13) n1⌢,m1(cid:13)f = kfk2L2(Rn+1)⊗L2(Rm+1)1 ⊗ 1,(cid:13)which i(cid:13)s a constant in L2 Rn+1 ⊗
L2 Rm+1 . (cid:0) (cid:1)
(cid:0) (cid:1)
Proof. Just exploit the full symmetry of f in the above definition of contractions.
(cid:3)
We are now ready to state the biproduct formula, which will be a crucial tool in
order to prove our main result.
Theorem3.5. Forn ,m ,n ,m N,letf L2 Rn1 L2 Rm1 =L2 Rn1+m1
1 1 2 2 ∈ ∈ + ⊗ + ∼ +
and g L2 Rn2 L2 Rm2 =L2 Rn2+m2 . Then it holds that
∈ + ⊗ + ∼ + (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
n1∧n2m1∧m2
p,r
(9) I I (f)♯I I (g)= I I f ⌢g .
n1 ⊗ m1 n2 ⊗ m2 n1+n2−2p⊗ m1+m2−2r
Xp=0 Xr=0 (cid:16) (cid:17)
Proof. Using a density argument together with the bisometry property of Wigner
bi-integrals,it is enough to prove the claim for functions f and g of the type a b
where a L2 Rn and b L2 Rm as the subset of functions ⊗
∈ + ∈ +
(cid:0) (cid:1) a b:(cid:0)a L(cid:1)2 Rn , b L2 Rm
⊗ ∈ + ∈ +
is dense in L2 Rn (cid:8)L2 Rm ). Le(cid:0)t the(cid:1)refore f (cid:0)= a(cid:1)(cid:9) b with a L2 Rn1 ,
+ ⊗ + ⊗ ∈ +
b L2 Rm1 and g =c d with c L2 Rn2 , d L2 Rm2 . It holds that
∈ + (cid:0) (cid:1) ⊗ (cid:0) (cid:1) ∈ + ∈ + (cid:0) (cid:1)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
I I (a b)♯I I (c d)=I (a) I (b)♯I (c) I (d)
n1 ⊗ m1 ⊗ n2 ⊗ m2 ⊗ n1 ⊗ m1 n2 ⊗ m2
=I (a) I (c) I (d) I (b).
n1 · n2 ⊗ m1 · m2
FREE QUANTITATIVE FOURTH MOMENT THEOREMS 9
Using the usual multiplication formula for Wigner integrals on both sides of the
tensor product, we get
I (a) I (c) I (d) I (b)
n1 · n2 ⊗ m1 · m2
n1∧n2 m1∧m2
p r
= I a⌢c I d⌢b
n1+n2−2p !⊗ m1+m2−2r !
Xp=0 (cid:16) (cid:17) Xr=0 (cid:16) (cid:17)
n1∧n2m1∧m2
p r
= I I a⌢c d⌢b
n1+n2−2p⊗ m1+m2−2r ⊗
Xp=0 Xr=0 (cid:16)(cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
n1∧n2m1∧m2
p,r
= I I (a b)⌢(c d) ,
n1+n2−2p⊗ m1+m2−2r ⊗ ⊗
Xp=0 Xr=0 (cid:16) (cid:17)
where the last equality follows from the identity (8). (cid:3)
Remark 3.6.
1. By taking m =m =0, f =u 1 and g =v 1, we recoverthe usual product
1 2
⊗ ⊗
formula (3) for Wigner integrals.
2. Note that a similar version of the above biproduct formula also holds for the
usual tensor product (with a slightly different definition for the bicontractions).
Furthermore, using the same methodology, one could also define contractions
and product formulae for higher order tensors.
3.2. Quantitative Fourth Moment Theorems. We are now in the position of
stating the main result of this paper, namely a bound on the quantity appearing
in the right hand side of (7) in terms of the fourth moment, which then leads to a
quantitative Fourth Moment Theorem for multiple Wigner integrals.
Theorem 3.7. For n N, let F = I (f) be a Wigner integral of order n with
n
∈
f L2 Rn symmetric and such that f 2 =1. Then, it holds that
∈ + k kL2(Rn+)
(cid:0) (cid:1) 2
(10) ϕ ϕ N−1F ♯( F)∗ds 1 1 C ϕ F4 2 ,
⊗ (cid:12) R ∇s 0 ∇s − ⊗ (cid:12) ≤ n −
(cid:12)(cid:12)Z + (cid:0) (cid:1) (cid:12)(cid:12) (cid:16) (cid:0) (cid:1) (cid:17)
where Cn = n12m(cid:12)(cid:12)ax{Pn(⌊u0⌋),Pn(⌈u0⌉)} with (cid:12)(cid:12) 
1
P (u)= u2(n u+1) 2(n u)2+4(n u)+3 ,
n
3 − − −
1 r(cid:0)(n) 2n2+4n 3 (cid:1)
(11) u = 4(n+1) −
0 5 − √34 − √32r(n)
(cid:18) (cid:19)
and
r(n)= 3 4n3+12n2+5√2 4n4+16n3+20n2+8n+5+22n+14.
q
p
Proof. In the following we will use the shorthandf(k) to denote the function given
s
by
f(k)(x ,...,x )=f(x ,...,x ,s,x ,...,x ).
s 1 n−1 1 k−1 k+1 n
10 SOLESNEBOURGUINANDSIMONCAMPESE
Observe that
( F)♯( F)∗ds
s s
R ∇ ∇
Z +
n
∗
= I I f(k) ♯ I I f(q) ds
R k−1⊗ n−k s q−1⊗ n−q s
kX,q=1Z + (cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
n
= I I f(k) ♯I I f(q) ds,
R k−1⊗ n−k s q−1⊗ n−q s
kX,q=1Z + (cid:16) (cid:17) (cid:16) (cid:17)
wherethe lastequality followsfromthe full symmetryofthe function f. Using the
product formula for bi-integrals proven in Theorem 3.5 yields
( F)♯( F)∗ds
s s
R ∇ ∇
Z +
n (k∧q)−1n−(k∨q)
= I I f(k) p⌢,r f(q) ds,
R k+q−2p−2⊗ 2n−k−q−2r s s
kX,q=1Z + Xp=0 Xr=0 (cid:16) (cid:17)
and by a Fubini argument one gets
( F)♯( F)∗ds
s s
R ∇ ∇
Z +
n (k∧q)−1n−(k∨q)
= I I f(k) p⌢,rf(q)ds .
k+q−2p−2⊗ 2n−k−q−2r R s s !
k,q=1 p=0 r=0 Z +
X X X
Thefullsymmetryoff impliesthatf(k) =f(q) forany1 k,q n,whichtogether
s s
≤ ≤
with Lemma 3.4 yields f(k) p⌢,rf(q)ds=f p+⌢r+1f. Hence,
R s s
+
(12)
R
n (k∧q)−1n−(k∨q)
( F)♯( F)∗ds= I I f p+⌢r+1f .
s s k+q−2p−2 2n−k−q−2r
R ∇ ∇ ⊗
Z + kX,q=1 Xp=0 Xr=0 (cid:16) (cid:17)
p+r+1
Exactly those summands I I f ⌢ f for which k +q
k+q−2p−2 2n−k−q−2r
⊗ −
2p 2 = 0 and 2n k q 2r = 0 yield the co(cid:16)nstant ter(cid:17)m f 2 1 1
− − − − k kL2(Rn+) · ⊗
(i.e. a constant in L2(R ) L2(R )). These conditions, along with the ranges of
+ +
⊗
summation, imply that k = q and p+r+1 = n. Therefore, fixing k, for which
we have n possibilities, fixes the other three indices q,p and r to take the values
k, k 1 and n k, respectively. Recalling that f 2 = 1, (12) can thus be
− − k kL2(Rn+)
rewritten as
( F)♯( F)∗ds
s s
R ∇ ∇
Z +
n (k∧q)−1n−(k∨q)
=n 1 1+ 1
{n−1−p−r>0}
· ⊗
k,q=1 p=0 r=0
X X X
p+r+1
I I f ⌢ f ,
k+q−2p−2 2n−k−q−2r
× ⊗
(cid:16) (cid:17)