A Constructive Boolean Central Limit Theorem

Bollettino U. M. I. (8) 10-B (2007), 000-000

A Constructive Boolean Central Limit Theorem. ANIS BEN GHORBAL - VITONOFRIO CRISMALE - YUN GANG LU

Sunto. -- Si fornisce una costruzione dei processi di creazione, distruzione e numero sullo spazio di Fock Booleano a mezzo di un teorema di limite centrale quantistico partendo da processi di creazione, distruzione e numero con tempo discreto.

Summary. -- We give a construction of the creation, annihilation and number processes on the Boolean Fock space by means of a quantum central limit theorem starting from creation, annihilation and number processes with discrete time.

1. -- Introduction. In quantum probability the notion of independence is not unique. During the past years emerged various inequivalent kinds of non-commutative independence, that were studied in a unified abstract picture in many papers (see [15], [5], [13]). Furthermore one knows that generally any notion of non-commutative independence generates a central limit theorem. Also for this case a unified abstract approach has been made by Accardi, Hashimoto and Obata in [4] and by Speicher and von Waldenfels in [16], where non-commutative general central limits are proved. More recently the interacting Fock space structure has suggested an idea to obtain constructive central limit theorems and this led to prove in [3] that any mean zero measure on the real line with finite moments of any order is the central limit in the sense of moments of a sequence of processes, satisfying rather weak conditions, on 1-mode type interacting Fock space. Furthermore Accardi and Bach showed in [1] that creation, annihilation and number operators in Boson Fock space without test function, can be obtained as central limit of operators in toy Fock spaces. This is possible also for fermionic, free and monotone Fock space (see [8], [10], [11]). One wish naturally to find a similar central limit theorem also for the boolean independence and Boolean Fock space and to this aim is devoted the present paper, where, after pointing out that Boolean Fock space can be seen as a 1-mode type interacting Fock space, we use some useful techniques of such a space in order to reach this goal.

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ANIS BEN GHORBAL - VITONOFRIO CRISMALE - YUN GANG LU

The notion of Boolean independence was introduced in [18] by W. von Waldenfels (namely ``interval partition'') and by Bozejko in [7] in the theory of the free groups and successively Speicher and Woroudi in [17] defined the Boolean convolution. This kind of non-commutative independence has been used also by Lenczewski in [9] and by Skeide in [14] for the theory of quantum stochastic calculus in the framework of Hilbert modules. More recently a quantum stochastic calculus on Boolean Fock space has been developed by Ben Ghorbal and Schu Èrmann in [6]. The paper is organized as follows: in Section 2 we define the Boolean Fock space, the simplest quantum probability model - quantum Bernoulli process - and proper constructive quantum stochastic processes (namely creation - annihilation - number processes) on the Boolean Fock space and with discrete time. In Section 3 we give results on the moments of such operator processes with respect to the ``vacuum'' state. In section 4 we present our main result: a quantum central limit argument is used to get the creation - annihilation - number processes on the Boolean Fock space starting from discrete-time processes. It is worth to mention that the family of processes used in our central limit theorem satisfies the singleton condition with respect to the vacuum state and the boundedness of the mixed moments (see [4] for more details). Then, as a consequence of [4], Lemma 2.4, only the pair partitions survive in the limit. Our constructive approach shows that discrete-time processes give exactly one pair partition, i.e. the interval partition. Finally we want to point out the differences between [16] and our paper:  we give a concrete construction rather than an abstract and general approach of existence;  in [16] the *-algebra and the state are the same either for the convergent sequence of the quantum random variables
N or their resulting limit, whereas our ; but in the limit one finds operators on sequence is made of operators on R2 the Boolean Fock space;  in [16] the GNS representation for the limit state has a Fock like structure on which the elements act as creation and annihilation operators, instead in our case the constructive approach allows to find also the number operator either in the convergent sequence or in the limit.

2. -- Notations and definitions. We firstly introduce some definitions and notations on Boolean Fock space. To this goal we consider an Hilbert space H: The Hilbert space G …H† :ˆ C  H

A CONSTRUCTIVE BOOLEAN CENTRAL LIMIT THEOREM

3

is called the Boolean Fock space over H: We denote by F the vector F :ˆ 1  0 which is called the vacuum vector in G …H† and define the vacuum expectation h
i as the state h
i :ˆ hF;
Fi In the following definitions the three basic operators of creation, annihilation and preservation on G …H† are presented in the usual way. DEFINITION 2.1. -- For any f 2 H and for any a 2 C; g 2 H the creation operator A‡ … f †…a  g† :ˆ 0  af is defined as a linear operator A‡ … f † : G …H† ! G …H† and has an adjoint A… f † : G …H† ! G …H† called the annihilation operator. REMARK 2.1. -- For any f 2 H and for any a 2 C; g 2 H A… f †…a  g† :ˆ h f ; gi  0 REMARK 2.2. -- For any f ; g 2 H h A… f †A‡ …g†i ˆ h f ; gi DEFINITION 2.2. -- For any B 2 B…H† and for any a 2 C; g 2 H we define the preservation (number) operator L…B† : G …H† ! G …H† such that L…B†…a  g† :ˆ 0  B…g† For any e 2 f 1; 0; 1g; for any B 2 B…H† and 8 ‡ > < A … f † if e A … f ; B† :ˆ L…B† if > : A… f † if

f 2 H we will denote eˆ1 eˆ0 eˆ

1

In order to state our main result we need some further definitions and notations from  starting   the  quantum Bernoulli process as in [8, 12]. Let us denote 1 0 e1 :ˆ ; e2 :ˆ : Clearly fe1 ; e2 g gives the usual base of R2 : For any 0 1 N  1; 0  n  N and 1  k1 < k2 < . . . < kn  N, we define ( …N † if n ˆ 0 d :ˆ e N …N† …2:1† d…k1 ;...;kn † :ˆ ;…k1 1† 1

…k2 k1 1†

… N kn † e1

e2 e1

e2 . . . e2 e1 if 1n N where hereinafter, when n ˆ 0; …k1 ; . . . ; kn † is understood as ;. We notice that

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ANIS BEN GHORBAL - VITONOFRIO CRISMALE - YUN GANG LU

the family n …N † …N † d…k1 ;...;kn † d…h1 ;...;hm † : 0  n; m  N; 1  k1 < . . . < kn  N; o 1  h1 < . . . < hm  N 2 2 is a base of the space M N 2 , where M2 :ˆ R R is the space of 2  2-matrices. The definition (2.1) can be generalized by inserting suitable ``test functions''. Hence we consider the space of all complex Riemann integrable functions L…‰0; 1Š† and for 1  n  N; 1  k1 < . . . < kn  N; f1 ; . . . ; fn 2 L2 …‰0; 1Š; dx†; define   nY1  kn kh …N † …N † d…k1 ;...;kn † … f1 ; . . . ; fn † :ˆ fn fh
d…k1 ;...;kn † N hˆ1 kh‡1

For any k ˆ 1; . . . ; N; for any f 2 L…‰0; 1Š†; for any 0  n 
N and 2 N into 1  k1 < . . . < kn  N we define a linear operator T ‡ N … f ; k† from R itself as follows ( …N † h i d…k† … f † if n ˆ 0 …N † ‡ T N … f ; k† d…k1 ;...;kn † :ˆ 0 if n  1 and we denote by T N … f ; k† the adjoint of T ‡ N … f ; k†: The following result allows us to compute the value of T N … f ; k†: LEMMA 2.1. -- For any k ˆ 1; . . . ; N; for any f 2 L…‰0; 1Š†; for any 0  n  N and 1  k1 < . . . < kn  N; ( …N †
h i if n ˆ 1; k ˆ k1 d; f …N † T N … f ; k† d…k1 ;...;kn † ˆ …2:2† 0 otherwise …N †

where d;


f :ˆ f

k N




…N †


d; :

PROOF. -- We firstly observe that for any n; m 2 N; 1  k1 < . . . < kn  N; 1  h1 < . . . < hm  N n D E Y k …N † …N † dhjj d…h1 ;...;hm † ; d…k1 ;...;kn † ˆ dnm jˆ1

where dnm is the Kronecker symbol. Hence we have that D

h iE D h i E …N † …N † …N † …N † … f ; k † d d…h1 ;...;hm † ; T N … f ; k† d…k1 ;...;kn † ˆ T ‡ ; d N …h1 ;...;hm † …k1 ;...;kn †

A CONSTRUCTIVE BOOLEAN CENTRAL LIMIT THEOREM

5

8  D E …N † …N † < N e T N … f ; k; B† :ˆ LN …B† > > : T N … f ; k†

if e ˆ 1 if e ˆ 0 if e ˆ

1

3. -- Moments of operators in discrete and Boolean case. In our main result we need to know the value of the following joint expectations D E …N † …N† d; ; T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d; hAen … fn ; Bn †


Ae1 … f1 ; B1 †i To this goal we present the following results for the discrete case. We observe that they hold in the Boolean case too. Their validity can be obtained just by …N † replying all the cases H to L…‰0; 1Š†; Ae to T eN and F to d; : LEMMA 3.1. -- For any n 2 N and e belongs to the set f 1; 0; 1gn :ˆ fe ˆ …e…n†; . . . ; e…1†† : e…i† 2 f 1; 0; 1g; 8i ˆ 1; . . . ; ng  i) if among T eNn … fn ; kn ; Bn †; . . . ; T eN1 … f1 ; k1 ; B1 † there are more annihilators than creators, then …N †

T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d;

ˆ0

ii) if the cardinality of the set fi : e…i† ˆ  1g is odd, then D E …N † …N † d; ; T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d; ˆ0 iii) if either e…n† 2 f0; 1g or e…1† 2 f 1; 0g, the scalar product D E …N † …N† d; ; T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d; is equal to zero.

A CONSTRUCTIVE BOOLEAN CENTRAL LIMIT THEOREM

7

 iv) if among T eNn … fn ; kn ; Bn †; . . . ; T eN1 … f1 ; k1 ; B1 † on has the same number of annihilators and creators, then there exists a constant c such that …N †

T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d;

…3:1†

…N†

ˆ cd;

PROOF. -- The results above follow directly from (2.2), (2.3), (2.4).

p

As in [3] we introduce the depth function of a given sequence e 2 f 1; 0; 1gn : DEFINITION 3.1. -- For n 2 N and e ˆ …e…n†;


; e…1†† 2 f 1; 0; 1gn we define the depth function (of the string e) de : f1; :::; ng ! f0;  1; :::;  ng by de … j † ˆ

j X

e… k †

kˆ1

ˆ jfe…k† : e…k† ˆ 1; k < jgj

jfe…k† : e…k† ˆ

1; k < jgj

Hence de … j† gives the relative number of creators (annihilators if negative) in the product
T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 † which are on the right side of e T Nj fj ; kj ; Bj : DEFINITION 3.2. -- f 1; 0; 1gn‡ is defined as the totality of all f 1; 0; 1gn satisfying the following conditions: n P e…k† ˆ de …n† ˆ 0; i) kˆ1

ii) e…1† ˆ 1 and e…n† ˆ

1;

iii) for all i ˆ 1; . . . ; n; de …i†  0; We denote by f 1; 0; 1gn‡;B the set f 1; 0; 1gn‡ such that the following condition holds: for any j ˆ 1; . . . ; n; if e… j† ˆ 1 (respectively e… j† ˆ 1) then e… j ‡ 1† 6ˆ 1 (respectively e… j ‡ 1† 6ˆ 1). LEMMA 3.2. -- For any n 2 N; and e 2 f 1; 0; 1gn ; the scalar product D E …N † …N † d; ; T eNn … fn ; kn ; Bn †


T eN1 … f1 ; k1 ; B1 †d; …3:2† can be nonzero only if e 2 f 1; 0; 1gn‡;B : PROOF. -- The conditions fulfilling Definition 3.2 can be obtained by using the same arguments developed in the proof of Lemma 3.2 in [3]. In order to prove that (3.2) does not vanish only if e 2 f 1; 0; 1gn‡;B ; we firstly
N observe that if e…2† ˆ 1; , then by definition of the creation operator on R2

8

…3:3†

ANIS BEN GHORBAL - VITONOFRIO CRISMALE - YUN GANG LU

D E …N † …N † ‡ d; ; T eNn … fn ; kn ; Bn †


T ‡ … f ; k †T … f ; k †d 2 2 1 1 N N ; D E …N † …N † ˆ d; ; T eNn … fn ; kn ; Bn †


T ‡ N … f2 ; k2 †d…k† … f † ˆ 0

Now we suppose that for all h  j 1, if e…h† ˆ 1; then e…h ‡ 1† 6ˆ 1 and prove it for h ˆ j: If e… j ‡ 1† ˆ 1; we have to compute D E
‡
…N † …N † ‡ d; ; T eNn … fn ; kn ; Bn †


T ‡ N fj‡1 ; kj‡1 T N fj ; kj


T N … f2 ; k2 †T N … f1 ; k1 †d;
After the reduction given by (2.4), on the right side of T ‡ N fj ; kj one finds only a sequence of creators and annihilators. Furthermore, from our hypothesis, the number of creators and annihilators there, must be equal. So, from (3.1) it follows that there exists a constant c such that the quantity above becomes equal to D
‡
…N † E …N † c d; ; T eNn … fn ; kn ; Bn †


T ‡ N fj‡1 ; kj‡1 T N fj ; kj d; thus giving zero as in (3.3).

p

As a consequence of Lemma 3.1, the scalar product (3.2) is non zero only if the number of creation and annihilation operators is the same. Hence hereinafter, if m is the cardinality of the operators in (3.2), then m ˆ 2n ‡ j, where n is the cardinality of creators (equivalently annihilators) and j the cardinality of preservation operators. Furthermore we denote V :ˆ fi ˆ 1; . . . ; m : e…i† ˆ 0 g (so is the cardinality of the set), put V ˆ: i1 ; . . . ij and jV j ˆ j, where as usual j
j  fh1 ; . . . h2n g :ˆ f1; . . . ; mgn i1 ; . . . ij : The following result states that any scalar product of the form D E …N † …N† d; ; T eNm … fm ; km ; Bm †


T eN1 … f1 ; k1 ; B1 †d; can be reduced to a product of factors, each of them formed by a scalar product containing exactly one annihilator and its following creator, in other words the Boolean condition of independence is here satisfied. LEMMA 3.3. -- For any m 2 N; for any e 2 f 1; 0; 1gm f ; . . . ; fm 2 ‡;B ; for any
1 2 L…‰0; 1Š†, 1  k1 < . . . < km  N and B1 ; . . . ; Bm 2 B L …‰0; 1Š; dx† D E …N † …N † d; ; T eNm … fm ; km ; Bm †


T eN1 … f1 ; k1 ; B1 †d; …3:4† 2n D     E Y …N † …N † ˆ d; ; T N fhp‡1 ; khp‡1 T ‡ N BVhp fhp ; khp d; pˆ1 p22N‡1

A CONSTRUCTIVE BOOLEAN CENTRAL LIMIT THEOREM

9

where for any p ˆ 1; . . . ; 2n; p 2 2N ‡ 1;  Vhp :ˆ ir ; . . . ; iq 2 V : hp‡1 < ir < . . . < iq < hp and BVhp :ˆ Bir


Biq ; ir ; . . . ; iq 2 Vhp : PROOF. -- In the notations introduced above, let h1 :ˆ 1, let hl and i1 be the indices relative respectively to the first annihilator and to the first preservation operator from the right in the left hand side of (3.4). From Lemma 3.2 we have that the left hand side of (3.4) is equal to D

eh …N † d; ; T N … fm ; km †


T Nl‡1 fhl‡1 ; khl‡1 ; Bhl‡1 T N fhl ; khl LN …Bir †


E
…N †


LN Bi1 T ‡ … f ; k †d N 1 1 ; Proposition 2.1 implies that the quantity above can be reduced to D


…N † E eh …N † d; ; T N … fm ; km †


T Nl‡1 fhl‡1 ; khl‡1 ; Bhl‡1 T N fhl ; khl


T ‡ N Bi1 f1 ; k1 d; and, by using repeatedly Lemma 3.2, and Proposition 2.1, we obtain D

eh …N † d; ; T N … fm ; km †


T Nl‡1 fhl‡1 ; khl‡1 ; Bhl‡1 T N fhl ; khl LN …Bir †

…N† E


LN Bi2 T ‡ N Bi1 f1 ; k1 d; D
eh …N † ˆ d; ; T N … fm ; km †


T Nl‡1 fhl‡1 ; khl‡1 ; Bhl‡1

…N † E
T N fhl ; khl T ‡ N Bir


Bi2 Bi1 f1 ; k1 d; Thus, necessarily l ˆ 2 and consequently Bir


Bi2 Bi1 ˆ BVh1 : The scalar product above can be rewritten as D  E
…N† ED …N†
‡ …N † …N † d T d ; ; T N … f m ; km †


T ‡ f ; k ; T f ; k B f ; k d d 1 1 N V h h h h 3 3 2 2 h1 N N ; ; ; Iterating the same procedure for any p ˆ 3; . . . ; 2n, p 2 2N ‡ 1; the thesis follows. p

4. -- Boolean Central Limit Theorem. In this section we present the main result of the paper. From now on H :ˆ L2 …‰0; 1Š; dx†: In order to prove our central limit theorem, as in [3] and [2], we consider for any e 2 f 1; 0; 1g; for any B 2 B…H†; for any 1  k  N; and f 2 L…‰0; 1Š† the ``normalized family''

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ANIS BEN GHORBAL - VITONOFRIO CRISMALE - YUN GANG LU

8 ‡ T … f ; k† > > < N e e … f ; k; B† :ˆ ck LN …B† T N > > : T N … f ; k†

if e ˆ 1 if e ˆ 0 if e ˆ

1

where fck g is a bounded sequence in R satisfying N 1 X lim p ck ˆ 1 N!1 N kˆ1

…4:1†

For any e 2 f 1; 0; 1g; we introduce the centered sum N 1 X e e … f ; k; B† SeN … f ; B† :ˆ p T N N kˆ1

and state the following central limit theorem. THEOREM 4.1. -- For any m 2 N; for any e 2 f 1; 0; 1gm ; for any f1 ; . . . ; fm 2 L…‰0; 1Š†, and B1 ; . . . ; Bm 2 B…H†; the limit, for N ! 1, of D E …N † …N † d; ; SeNm … fm ; Bm †


SeN1 … f1 ; B1 †d; …4:2† m is equal to zero whenever e 2 f 1; 0; 1gm nf 1; 0; 1gm ‡;B and if e 2 f 1; 0; 1g‡;B it is equal to

hAem … fm ; Bm †


Ae1 … f1 ; B1 †i PROOF. -- In fact (4.2) can be rewritten as 1 m N2

N D E X … N † e em …N † e e1 d; ; T N … fm ; km ; Bm †


T N … f1 ; k1 ; B1 †d;

k1 ;...;km ˆ1

hence obtaining the first part of the thesis. If e 2 f 1; 0; 1gm ‡;B , then, from Lemma 3.3, this quantity is equal to 2 0 13 j N Y X 1 4 j @ ckiq A5 N 2 qˆ1 kiq ˆ1 2

3 2n N D     E Y X 6 1 …N † …N † 7 d ; T N fhp‡1 ; khp‡1 T ‡ 4 n 5 N BVhp fhp ; khp d; N pˆ1 k ˆ1 ; p22N‡1

hp

where m ˆ 2n ‡ j and, as defined above, 2n and j are respectively the cardinality

11

A CONSTRUCTIVE BOOLEAN CENTRAL LIMIT THEOREM

of annihilator-creators and preservation operators. By definition of discrete annihilations, creation and preservation operators, we obtain 2 0 2 3     j N 2n N Y X Y X k k hp hp 7 61 B 6 1 ckiq †Š
4 n f hp‡1 BVhp fhp 4 j @ 5 N N N 2 N qˆ1 kiq ˆ1 pˆ1 khp ˆ1 p22N‡1

2

0 13 2 3     j N N X Y X 1 1 k k h h 1 1 @ 5 ckiq A5
4 f B Vh 1 f h 1 ˆ4 j N k ˆ1 h2 N N N 2 qˆ1 kiq ˆ1 h1 2 1 ...  4 Nk



N X h2n 1 ˆ1

f h2n

kh2n N

 1

BVh2n 1 fh2n

3   kh2n 1 5 1 N

Taking the limit for N ! 1, the first factor converges to 1; thanks to (4.1), while, for the remaining others, we recognize in each of them, a Riemann-Lebesgue sum. Hence we have 0 1 1 2n 2n D E Y Y @ q f h …x†BV fh …x†dxA ˆ fhp‡1 ; BVhp fhp p hp p‡1 pˆ1 p22N‡1

0

pˆ1 p22N‡1

H

By definition of annihilation, creation and preservation operators, the quantity above is equal to 2n D  2n D       E Y Y
E A fhp‡1 L BVhp A‡ fhp ˆ A fhp‡1 A‡ BVhp fhp

pˆ1 p22N‡1

pˆ1 p22N‡1

By applying the Boolean counterpart of Lemma 3.3, we finally obtain hAem … fm ; Bm †


Ae1 … f1 ; B1 †i

p

Acknowledgment. A. B. G. was supported by QP-Applications, European Network Training Research ``Quantum Probability with Applications to Physics; Information Theory and Biology'', Contract No. HPRN-CT-2002-00279. V. C. thanks Drs. A. D. Krystek and L. J. Wojakowski for useful suggestions. REFERENCES [1] L. ACCARDI - A. BACH, The harmonic oscillator as quantum central limit theorem for monotone noise, preprint (1985). [2] L. ACCARDI - A. BEN GHORBAL - V. CRISMALE - Y. G. LU, Projective independence and qunatum central limit theorem, preprint (2007).

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[3] L. ACCARDI - V. CRISMALE - Y. G. LU, Constructive universal central limit theorems based on interacting Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 8, no. 4 (2005), 631-650. [4] L. ACCARDI - Y. HASHIMOTO - N. OBATA, Notions of independence related to the free group, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 1, no. 2 (1998), 201-220. [5] A. BEN GHORBAL - M. SCHUÈRMANN, Noncommutative notions of stochastic independence, Math. Proc. Cambridge Philos. Soc., 133, no. 3 (2002), 531-561. [6] A. BEN GHORBAL - M. SCHUÈRMANN, Quantum stochastic calculus on Boolean Fock Space, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7, no. 4 (2004), 631-650. [7] M. BOZEJKO, Uniformly bounded representations of free groups, J. Reine Angew. Math., 377 (1987), 170-186. [8] M. DE GIOSA - Y. G. LU, The free creation and annihilation operators as the central limit of the quantum Bernoulli process, Random Oper. Stoch. Eq. 5, no. 3 (1997), 227236. [9] R. LENCZEWSKI, Stochastic calculus on multiple symmetric Fock spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 4, no. 2 (2001), 309-346. [10] V. LIEBSHER, On a central limit theorem for monotone noise, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2, no. 2 (1999), 155-167. [11] Y.G. LU, The Boson and Fermion quantum Brownian motions as quantum central limits of the quantum Bernoulli processes, Boll. Un. Mat. Ital. B (7) 6, no. 2 (1992), 245-273. [12] P.A. MEYER, ``Quantum probability for probabilists'', Lecture Notes in Mathematics 1538, Springer-Verlag, Berlin, 1993. [13] N. MURAKI, The five independencies as natural products, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 6, no. 3 (2003), 337-371. [14] M. SKEIDE, Quantum stochastic calculus on full Fock modules, J. Funct. Anal., 173, no. 2 (2000), 401-452. [15] R. SPEICHER, On universal product, Fields Inst. Commun., 12 (1997), 257-266. [16] R. SPEICHER - W. VON WALDENFELS, A general central limit theorem and invariance principle, in Quantum Probability and Related Topics, Editor L. Accardi, World Scientific Vol., 9 (1994), 371-387. [17] R. SPEICHER - R. WOROUDI, Boolean convolution, Fields Inst. Commun., 12 (1997), 267-279. [18] W. VON WALDENFELS, An approach to the theory of pressure broadening of spectral lines, In: ``Probability and Information Theory II'', M. Behara - K. Krickeberg - J. Wolfowitz (eds.), Lect. Notes Math., 296 (1973), 19-64. Anis Ben Ghorbal: Centro Internazionale Vito Volterra, UniversitaÁ di Roma ``Tor Vergata'', Via Columbia 2, I-00133 Roma, Italy e-mail: [email protected] Vitonofrio Crismale - Yun Gang Lu: Dipartimento di Matematica, UniversitaÁ degli Studi di Bari, Via E. Orabona 4, I-70125 Bari, Italy e-mail: [email protected] [email protected] ________ Pervenuta in Redazione il 23 gennaio 2006