The Central Limit Theorem for Free Additive Convolution

Transcription

1 journal of functional analsis 140, (1996) article no The Central Limit Theorem for Free Additive Convolution Vittorino Pata*, - Dipartimento di Elettronica per l'automazione, Universita di Brescia, Ital Received Jul 25, 1995 Let [X n ] n=1 be a sequence of free, identicall distributed random variables with common distribution +. Then there exist sequences [B n ] n=1 and [A n] n=1 of positive and real numbers, respectivel, such that sequence of random variables X 1 +}}}+X n A n B n converges in distribution to the semicircle law if and onl if the function H + ( )= [, ] t 2 d+(t) is slowl varing in Karamata's sense. In other words, the free domain of attraction of the semicircle law coincides with the classical domain of attraction of the Gaussian. We prove an analogous result for normal domains of attraction in the sense of Linnik Academic Press, Inc. 1. INTRODUCTION Man results in recent ears have corroborated the idea that the theor of free random variables is a suitable framework for a noncommutative analogue of classical probabilit. The ke concept in this theor is the notion of freeness, which is one of the possible tpes of independence for noncommuting random variables. As in classical probabilit the concept of independence gives rise to the classical convolution, here the concept of freeness leads to a binar operation on the probabilit measures on the real line, namel, the free additive convolution. Given two probabilit measures on the real line, their free additive convolution is uniquel determined as the distribution of the sum of free real random variables having those laws. A surprising number of noncommutative counterparts of classical weak * The author was supported in part b the Consiglio Nazionale delle Ricerche, Ital. - vpatavmimat.mat.unimi.it Copright 1996 b Academic Press, Inc. All rights of reproduction in an form reserved.

2 360 VITTORINO PATA limit results hold for free additive convolution, for instance, the characterizations of infinitel divisible and stable laws [2, 3, 9, 10, 11, 12], the weak law of large numbers [1, 8], and the central limit theorem [9, 12]. Nonetheless we should stress that these results are not mere translations in the noncommutative settings of classical arguments, but involve genuinel new ideas and techniques. Following this pattern, in this paper we provide a characterization of the laws satisfing the central limit theorem completel analogous to the classical one. The quite striking consequence of this fact is that a law satisfies the free central limit theorem if and onl if it satisfies the classical central limit theorem, i.e., the domain of attraction of the normal law and the domain of attraction of the semicircle law (the noncommutative analogue of the normal law) coincide. 2. PRELIMINARIES AND NOTATION To fix terminolog, we recall in this section some definitions and results from [3]. A W*-probabilit space (A, {) consists of a complex noncommutative unital von Neumann algebra A with a normal faithful trace {. As usual we identif A with?(a), where? is a faithful representation of A on a Hilbert space. A (real) random variable is a selfadjoint operator X affiliated with A (i.e., the spectral projections of X belong to A). Thus it is possible to associate to a random variable X a probabilit measure + X on the real line R, called the distribution of X, b+ X ={be X, where E X is the spectral measure of X. Free additive convolution of probabilit measures on R was introduced in [12] (see also [15] for a detailed presentation of the subject). Namel, given two measures + and, there exist two free random variables (in the sense of [12]) X and Y such that +=+ X and =+ Y, and the free additive convolution of + and, denoted b +, is defined to be the distribution of X+Y. In the sequel, let C denote the complex field, C + and C the upper and lower half plane, respectivel, and set 1 : =[z=x+i # C : >0 and x <:], 1 :, ; =[z # 1 : : >;], where : and ; are positive numbers. Given a probabilit measure + on R, its Cauch transform G + : C + C is defined as the expectation of the resolvent of the operator X associated to +, to wit, G + (z)= + 1 zt d+(t), z # C+.

3 FREE ADDITIVE CONVOLUTION 361 The Cauch transform is analtic and lim z z # 1 : zg + (z)=1 for ever :>0. In fact this propert characterizes an analtic function G: C + C as the Cauch transform of a probabilit measure + on R. It is useful for our purposes to introduce the reciprocal of the Cauch transform, F + (z)=1g + (z), which maps C + to C +. From the characterization of G + (z), it is clear that lim z z # 1 : F + (z) =1 (2.1) z for ever :>0. B the Nevanlinna representation, an analtic function F: C + C + can be written as F(z)=a+bz+ + 1+tz tz d_(t), z # C+, for some positive finite measure _ on R, some a # R, and some b0. Thus from (2.1) the function F + has a Nevanlinna representation of the form F + (z)=a+z+ + 1+tz tz d_(t), z # C+. (2.2) for some positive finite measure _ on R, and some a # R. It is also clear from (2.2) that IF + (z)iz, z # C +. For ever :>0 there exists ;>0 such that F + has an inverse (relative to composition) F 1 + defined on 1 :, ;. Obviousl if 1 :, ; /domain (F 1 + ), then IF 1 + (z)iz, z # 1 :, ;. Thus it is possible to define a function, + : 1 :, ; C (with ;=;(:)), which is uniquel determined b the measure +, as, + (z)=f 1 + (z)z, z # 1 :, ;. The function, +, sometimes called the,-function of +, plas a central role in free probabilit, similar to the role plaed b the characteristic function in classical probabilit. Indeed, given two probabilit measures + 1 and + 2,

4 362 VITTORINO PATA we have, + =, +1 +, +2 if += This surprising result was first proved b Voiculescu [13] for compactl supported measures, and was later extended b Maassen [9] for measure of finite variance, and finall b Bercovici and Voiculescu [3] for general measures. The following proposition from [1, 3], which describes the dependence of, +,on+, will be a crucial tool for our purposes. Proposition 2.1. Let [+ n ] n=1 be a sequence of probabilit measures on R. The following assertions are equivalent: (i) the sequence [+ n ] n=1 converges weakl to a probabilit measure +; (ii) there exist :, ;>0 such that the sequence [, +n ] converges n=1 uniforml on the compact subsets of 1 :, ; to a function,, and, +n (z)=o( z ) uniforml in n as z, z # 1 :, ; ; (iii) there exist :$, ;$>0 such that the functions, +n are defined on 1 :$, ;$ for ever n, lim n, +n (i) exists for ever >;$, and, +n (i)=o() uniforml in n as. Moreover, if (i) and (ii) are satisfied, we that,=, + in 1 :, ;. 3. SLOWLY VARYING FUNCTIONS Let us first introduce some notation. Let f and h (#R or C) depend on a parameter x which tends, sa, to a. Then we write f th ( f=o(h)) as xa, iffh1(fh0) as xa. Definition 3.1. A measurable function l: [0, +)[0, +] is said to be slowl varing (in Karamata's sense) if there exists 0 0 such that l(( 0,+))/(0, +), and for ever t>0, l(t) lim l( ) =1. A measurable function r: [0,+)[0, +] is said to be regularl varing of index \ if it is of the form r( )= \ l(), with l slowl varing. We denote b R \ the class of regularl varing functions of index \ (R 0 is the class of slowl varing functions). Analogousl, a positive sequence [h n ] n=1 is said to be slowl varing if, for ever k=1, 2, 3,..., h lim kn =1. h n

5 FREE ADDITIVE CONVOLUTION 363 Not all slowl varing sequences extend to slowl varing functions; however, if h n is a slowl varing sequence such that h n+1 h n 1, then there exists l # R 0 such that l(n)=h n, moreover, l can be chosen to be continuous (see [5], Proposition and Theorem 1.9.8). It is eas to see that if l # R 0, the following hold: (i) (l()) : # R 0 for all : # R; (ii) l(+t)tl() asfor all t0; (iii) For all =>0 lim = l( )=, lim = l( )=0. We now recall some remarkable properties of this class of functions (for a complete treatise on the subject see [5]). P1. Representation theorem. A function l=[0, +)[0, +] belongs to R 0 if and onl if it is of the form l( )=c() exp \ x =(t) dt + t ( x), for some x>0, where c( ) and =( ) are measurable, c( ) c # (0, +), =( ) 0as. P2. If l#r 0 it is possible to find a positive sequence [B n ] such n=1 that B n as n, and for ever >0, n B 2 n l(b n )t1 (n). P3. Uniform convergence. If l#r 0 then l(t)tl( ) ( ) uniforml in t on each compact set K/(0, +). P4. If l#r 0, x is so large that l( ) is locall bounded in [x, +) and :>1, then x P5. If l#r 0, :<1, then t : l(t) dtt 1 :+1 :+1 l() (). lim + t : l(t) dt<,

6 364 VITTORINO PATA and + t : l(t) dtt 1 :+1 :+1 l() (). P6. Asmptotic inversion. If f#r \ with \>0, there exists g # R 1\ with f(g( ))tg( f())t (). We would just like to sketch the proof of (P2) which we did not find in the literature. Indeed, b Proposition in [5], we can assume that l is continuous. So (P2) is a direct consequence of the fact that 2 l( ) 0as. In the sequel we will use the following notation. For x>0, we denote I =[, ], 2 =(,)_(,+), and 2, x =(x,)_(,x). Further, given a positive measure + on R and :0, we define + (:) = + t : d+(t), provided that the above integral exists, i.e., converges absolutel (we sometimes write with abuse of notation + (:) < to mean that the integral exists). We also denote b Var[+] the variance of +. Definition 3.2. For a finite positive measure + on R we introduce the functions / +, H + and L + :[0,+)[0, +) b / + ()= 2 +(2 ), H + ( )= I t 2 d+(t), and L + ( )= + 2 t 2 2 +t 2 d+(t). Notice that H + ( ), L + ( )>0 for large enough if and onl if +{:$ 0, :0, where $ 0 is the Dirac measure at zero. Moreover it is clear that H + ( ) and L + ( ) are increasing in, and using the monotone convergence theorem and the dominated convergence theorem, it is eas to see that H + ( ) and L + ( ) are bounded if and onl if + has finite variance; in which case H + ( ), L + ( ) + (2) as. It is a well-known fact that H + # R 0 if and onl if / + ( )=o(h + ()) as (see, e.g., Theorem VIII.9.2 in [6]).

7 FREE ADDITIVE CONVOLUTION 365 Proposition 3.3. Given a finite positive measure + on R, H + # R 0 if and onl if L + # R 0. In this case H + ( )tl + ( ) as. Proof. It is clear that L + ( )H + ()+/ + (). Suppose now that either H + or L + is slowl varing. Then for ever n1, L + () H + () =C n() n H + ()\ 2 2 t 2 I n 2 2 +t d+(t)+ n 2 2 t n 2 2 +t d+(t) +, 2 where C n ( )t1 as. Now n 2 2 t 2 I n 2 2 +t d+(t) n2 2 n 2 +1 H +(), moreover, integrating b parts, we get 2 n 2 2 t 2 n 2 2 +t 2 d+(t) Therefore = lim x 2, x n 2 2 t 2 n 2 2 +t 2 d+(t) = lim x \ n2 n 2 +1 / +() n2 2 n 2 2 +x / +(x)+ x 2 = n2 n 2 +1 / +()+ + n2 n 2 +1 / +(). from where we get Hence C n ( ) n 2 lim inf n n 4 4 t(n 2 2 +t 2 ) 2 / +(t) dt n 2 n 2 +1\ 1+ / +() H + ()+ L +() H + (), n 2 C n ( ) n 2 +1 L +() H + ()+/ + () 1. L + ( ) lim sup H + ( )+/ + () 2n 4 4 t(n 2 2 +t 2 ) / +(t) dt + 2 L + ( ) H + ( )+/ + () 1.

8 366 VITTORINO PATA Letting n we conclude that L + ( )th + ( )+/ + () (). (3.1) If H + ( )#R 0 the result follows at once from (3.1), since in that case / + ( )=o(h + ()) as. Assume then that L + # R 0. The equalit H + ( )+/ + ()= + f (t) d+(t), where if t, f (t)= {t2 2 if t >, leads to (H + (2)+/ + (2))(H + ()+/ + ()) = [2, ]_[,2] (t 2 2 )d+(t)+3 2 +(2 2 ) 3 2 +(2 2 ) = 3 4 / +(2). B (3.1), dividing both sides of the above equation b L + (2), we finall get that / + ( )=o(l + ()) as, which implies the result. K If H + # R 0 then + (:) < for all :<2 (cf. Theorem in [7]). Thus a direct consequence of Proposition 3.3 is the following result. Corollar 3.4. Let + be a finite positive measure on R. If L + # R 0, then + (:) < for all :<2. The next technical lemma will be needed to prove Proposition 3.6. Lemma 3.5. follows that Let + be a finite positive measure on R. Then if L + # R 0 it lim 1 L + ( ) + t 3 2 +t 2 d+(t)=0.

9 FREE ADDITIVE CONVOLUTION 367 Proof. If L + () is bounded + has finite variance, and the proof of the lemma is carried out b the dominated convergence theorem. Assume then that L + ( ) as. We have that } + t 3 2 +t 2 d+(t) } I t 3 2 +t 2 d+(t)+ 2 t 3 2 +t 2 d+(t) 1 I t 3 d+(t)+ 2 t d+(t). (3.2) Select =>0. Since b Proposition 3.3 / + ( )=o(l + ()) as, there exists n>0 such that / + ( )L + ( )<=4 for all n. For the first term of the right-hand side of (3.2), integration b parts ields B (P4), 1 I t 3 d+(t)=/ + ()+ 3 0 / + (t) dt 3 / + (t) dt. (3.3) 3 / + (t) dt= 3 0 n / + (t) dt+ 3 0 / + (t) dt n 3n3 + (0) + 3= 4 L + (t) dt n = 3n3 + (0) + 3= 4 C() L +(), where C( )t1 as. For the second term of the right-hand side of (3.2), again integrating b parts, 0 t d+(t)= lim 2 t d+(t) x 2, x = lim x \ / +( ) x / +(x)+ x =/ + ( )+ + / + (t) dt. t 2 / + (t) t 2 dt + B (P5), for n, + / + (t) t 2 dt = 4 + L + (t) dt= = t 2 4 K( ) L +( ),

10 368 VITTORINO PATA where K( )t1 as. Combining (3.3) and (3.4), and recalling that / + ( )=o(l + ()) as, lim sup 1 L + ( ) } + lim sup ==. t 3 2 +t d+(t) } 2 \ 3n 3 + (0) L + ( ) +3= 4 C( )+/ +() L + () += 4 K() + Letting = 0, we finish the proof. K We conclude this section with a proposition which provides a relation between a probabilit measure + on R and the measure _ that appears in the Nevanlinna representation of F +. We sa that a probabilit measure + on R is non-degenerate if +{$ c, c # R, where $ c is the Dirac measure at c. Proposition 3.6. Let + be a non-degenerate probabilit measure on R whose reciprocal of the Cauch transform is given b (2.2). Then H + # R 0 if and onl if either H _ # R 0 or H _ =0. In this case the following hold: and + (1) =_ (1) a; (3.5) H + ( )(+ (1) ) 2 th _ ()+_ (0) (). (3.6) Proof. The fact that + is non-degenerate implies that _ is not the zero measure. Suppose first that H + # R 0 (in particular this implies that + has finite mean). B Proposition 3.3, IG + (i)= + 2 +t 2 d+(t) = L +() = H +()+ 1 3 o(h +()) ().

11 FREE ADDITIVE CONVOLUTION 369 Moreover b Lemma 3.5 RG + (i)= + 2 +t 2 d+(t) = +(1) t 3 2 +t 2 d+(t) = +(1) o(h +()) (). Thus G + (i) 2 = 1 2+(+(1) ) 2 2H + () o(h +()) (). So, defining V + ( )=H + ()(+ (1) ) 2 (notice that V + # R 0 as well), we have the following estimates: Since and IF + (i)= IG +(i) G + (i) 2 =+1 V +()+ 1 o(v +()) (), (3.7) RF + (i)= RG +(i) G + (i) 2=+(1) + 1 o(h +()) (). (3.8) from (3.7) we get IF + (i)=+ 1 L _()+ + lim + lim 2 +t 2 d_(t), 2 2 +t 2 d_(t)=_(0), V + ( ) L _ ( )+_ (0)=1. Hence either L _ # R 0 or L _ ( )=0 for ever, and b Proposition 3.3, we conclude that V + ( )th _ ( )+_ (0) as (in particular, if L _ =0, + has finite variance, and Var[+]=_ (0) ). Finall (3.8) ields + (1) =a+ lim + (1 2 )t d_(t)=a_ (1) 2 +t 2

12 370 VITTORINO PATA b the dominated convergence theorem. Conversel, suppose that H _ # R 0 (and so b Proposition 3.3 L _ # R 0 ) or H _ =0 (and so L _ =0). Then, defining a~ = a _ (1), F + (z)=a~ + z t 2 tz d_(t), z # C+. Thus, IF + (i)=+ + (1+t 2 ) 2 +t 2 d_(t)=+_(0) +L _ () + 1 o(1) (), and, b the dominated convergence theorem, RF + (i)=a~ t + t t 2 d_(t)=a~ + 1 o (1) (). Thus F + (i) 2 = 2 +2_ (0) +2L _ ()+a~ 2 + o (1) (), and we get IG + (i)= IF +(i) +_(0) +a~ 2 + L F + (i) 2=1 _ ( ) + 1 o (1) (), 3 3 which implies 3 \ IG +(i)+ 1 + =L +()t_ (0) +a~ 2 + L _ ( ) (). So L + # R 0, and the result follows from Proposition 3.3. The following corollar, which extends Proposition 2.2 in [9], is an immediate consequence of Proposition 3.6. K Corollar 3.7. Let + be a probabilit measure on R whose reciprocal of the Cauch transform is given b (2.2). Then + has finite variance if and onl if _ has finite variance. In this case + (1) =_ (1) a, and Var[+]= _ (0) +_ (2). Proof. If +=$ c then _ is the zero measure, a=c, and the result is obvious. If + is non-degenerate, the result comes directl from (3.5), and appling the monotone convergence theorem to (3.6). K

13 FREE ADDITIVE CONVOLUTION THE CENTRAL LIMIT THEOREM The central limit theorem for free additive convolution was first proved in [12] b Voiculescu for compactl supported measures. In the identicall distributed situation, the theorem states that, given a sequence [X n ] n=1 of free identicall distributed random variables with a compactl supported common distribution + with standard deviation b>0, the random variable (X 1 + (1) )+ }}} +(X n + (1) ) - nb converges in distribution (as n ) to the so-called semicircle law # 0, 1. Actuall, in this compact support case a stronger form of convergence was shown to hold [4]. The semicircle law # a, b, for a0 and b>0 is defined as follows: 1 d# a, b 2?b (t)={ - 2 4b2 (ta) 2 dt if t #[a2b, a+2b], 0 otherwise. This result was extended b Maassen [9] to measures of finite variance. In [10] we proved the converse of this fact for a particular class of probabilit measures, precisel, -infinitel divisible measures (see [3]). The semicircle law was first encountered b Wigner [16], when he studied the distribution of the eigenvalues of large smmetric random matrices with independent Gaussian entries. An explanation of the appearance of the semicircle law in that situation was provided b Voiculescu [14]. Indeed, he showed that scaled sums of such matrices are asmptoticall free, and the semicircle law appears as a consequence of the central limit theorem. Thus the semicircle law seems to pla a universal role in free probabilit, similar to the role plaed b the normal law in classical probabilit. Notice also that, like the normal law, the semicircle law is completel determined b its mean and its variance. Indeed # (1) =a and Var[# a, b a, b]=b 2. A simple calculation shows that from where we get, #a, b (z)=a+ b2 z,, #a1, b 1 +, #a2, b 2 =, #a1 +a 2, - b b 2 2,

14 372 VITTORINO PATA which implies # a1, b 1 # a2, b 2 =# a1 +a 2, - b 2 1 +b2 2, (the similarit with the classical convolution of the corresponding normal laws is apparent). In order to state our results we need some preliminar definitions. Let [X n ] n=1 be a free sequence of random variables with common distribution + in a W*-probabilit space, and let [B n ] and n=1 [A n ] n=1 be sequences of positive and real numbers, respectivel. Set Z n =Z n (+, B n, A n )= X 1+}}}+X n B n A n. Notice that, since we are interested in the distribution of Z n, we do not emphasize the dependence of Z n on the particular choice of the sequence [X n ] n=1, which in fact does not affect + Z n. Definition 4.1. If for some choice of the constants B n and A n the random variable Z n (+, B n, A n ) converges in distribution to # 0, 1, then + belongs to the domain of attraction of the semicircle law (and we write + # D[# 0, 1 ]). Obviousl, + # D[# 0, 1 ] if and onl if + # D[# a, r ] for all a0 and r>0. Indeed, it is enough to perform an appropriate scaling of the constants B n and A n.if+#d[# 0, 1 ], then + is non-degenerate, and the constants B n must necessaril be of the form B n =- nh n, where [h n ] n=1 is a slowl varing sequence, and h n+1 h n t1asn(cf. Theorem 5.5 in [10]). An interesting case is when h n is constant. This leads to the following definition. Definition 4.2. If for some choice of the constants A n and for some positive constant b the random variable Z n (+, - nb,a n ) converges in distribution to # 0, 1, then + belongs to the normal domain of attraction of the semicircle law (and we write + # ND[# 0, 1 ]). Again, + # ND[# 0, 1 ] if and onl if + # ND[# a, r ] for all a0 and r>0. In [10] we proved that stable laws and onl stable laws (stable with respect to free additive convolution) have a non-empt domain of attraction. Here, using the techniques developed in [1], we want to characterize the domain of attraction of a particular stable law, namel, the semicircle law. This would provide the most general form of the central limit theorem for free additive convolution. In the classical case, a law + belongs to the domain of attraction of the normal law if and onl if + is nondegenerate, and has almost finite variance, i.e., the function H + # R 0 (see, e.g., [6, 7]).

15 FREE ADDITIVE CONVOLUTION 373 Our result is that, in analog with the classical case, + # D[# 0, 1 ] if and onl if + has almost finite variance, and + # ND[# 0, 1 ] if and onl if + has finite variance. In particular this implies that the free (normal) domain of attraction of the semicircle law coincide with the (normal) domain of attraction of the normal law. Before going an further, let us formulate the problem in terms of free additive convolution. Given a selfadjoint random variable X affiliated with some W*-tracial probabilit space, and a scalar c>0, we have + c X=D c + X, where the dilation D c of a measure + is defined b D c +(B)=+(c 1 B), for ever Borel subset B/R. This relation can be translated into, +cx (z)=c, +X\ z c+ at the level of,-functions. In view of the above considerations, our result can be stated as follows. Theorem 4.3 (General CLT). Let + be a non-degenerate probabilit measure on R. The following conditions are equivalent: (i) there esist sequences [B n ] and [A n=1 n] n=1 of positive and real numbers, respectivel, such that the sequence of measures [ n ] n=1 converges weakl to # 0, 1, where n =D 1Bn + }}}D 1Bn + $ An ; n times (ii) the function H + is slowl varing. Theorem 4.4 (CLT). Let + be a non-degenerate probabilit measure on R. The following conditions are equivalent: (i) there esist b>0 and a real sequence [A n ] n=1 such that the sequence of measures [ n ] converges weakl to # n=1 0, 1, where n =D 1- nb +}}}D 1-nb + $ An ; ntimes (ii) the measure + has finite variance.

16 374 VITTORINO PATA Moreover if (ii) is satisfied, the constants b and A n in (i) can be chosen to be and b=(var[+]) 12, A n = - n(1). b 5. PROOF OF THE MAIN RESULT We first establish some preliminar results. The proof of the following lemma is in [1]. Lemma 5.1. Let + be a probabilit measure on R. Given a truncated cone 1 :, ;, there exists a truncated cone 1 :$, ;$ such that F + (1 :$, ;$ )/1 :, ;. Lemma 5.2. Let + be a non-degenerate probabilit measure on R in the domain of attraction of the semicircle law. Then there exists a function l # R 0 such that IF + (i)=+ 1 l()+1 l() o(1) (). Proof. B Proposition 2.1 (ii), there exist a positive sequence [h n ] n=1 and a real sequence [A n ] n=1 such that, denoting b n the distribution of Z n (, - nh n,a n ), lim n, n (z)=, #0, 1 (z)= 1 z uniforml on the compact subsets of some truncated cone 1 :0, ; 0. But so, taking the imaginar part,, n (z)= -n h n, + (-nh n z)a n, I, + (- nh n z)= h n -n I \ 1 z+ + h n o(1) (n) (5.1) -n uniforml on the compact subsets of 1 :0, ; 0. Since [h n ] n=1 is a slowl varing sequence, and h n+1 h n t1asn, there exists a h # R 0 such that

17 FREE ADDITIVE CONVOLUTION 375 h(n)=h n. The function f( )=-h()#r 12, and b (P6) there exists a function g( )= 2 k(), with k # R 0, such that g( f( ))t as. Hence (h( )) 2 k(-h())t1 (). Define a function l # R 0 b l( )=(k()) 1.B(P3), l(-nh n z )t(h n ) 2 (n) (5.2) uniforml in z on ever compact subset of C +. B Lemma 5.1, there exist positive numbers : 2, : 3, ; 2, ; 3 (with : 0 >: 2 and ; 0 <; 2 ) such that F + has an inverse on 1 :2, ; 2, and 1 :0, ; 0 #1 :2, ; 2 #F + (1 :3, ; 3 ). Therefore for an z # 1 :3, ; 3 it follows that F + (z)=z, + (F + (z)). (5.3) In particular, defining : 1, ; 1 such that : 0 >: 1 >: 2 and ; 0 <; 1 <; 2, (5.1) holds for ever z in the compact set Observe that K=1 :1, ; 1 [ =!+i' : '2; 1 ].. n=1 - nh n K#1 :1,;$ 1 for some ;$ 1 >; 1. Indeed, - nh n as n, and 2 - nh n > -n+1 h n+1 for n large enough. Thus using (5.2), renaming - nh n zb z, and observing that for z # 1 :1, (5.1) becomes Iz 1 z 1-1+: 2 1 I, + (z)=i z+ \1 l( z )+ 1 l( z ) o(1) z as z, z # 1 :1. Since F + (i)ti as, b (P3) we conclude that l( F + (i) )=l( (1+o(1)) )tl() (), and therefore I, + (F + (i))= 1 l()+1 l() o(1) (). (5.4) Joining (5.3) and (5.4), we finall get IF + (i)=i, + (F + (i))=+ 1 l()+1 l() o(1) (). K

18 376 VITTORINO PATA The proof of the following lemma, which we include for reader's convenience, uses a standard argument of complex analsis. Lemma 5.3. Let + be a probabilit measure on R. Then as z is an truncated cone 1 :. d dz F 1 + (z)=1+o(1) Proof. For :>0, and z large enough, F 1 + (z)=z+ (z), with (z)= o( z ) as z, z # 1 2:. Then there exists ;>0 such that 1 :, ; /1 2:, ; / domain(f 1 + ). Defining =tan 1 (:) and =tan 1 (2:)tan 1 (:), and taking z # 1 : with z large enough such that z (cos()sin())>; (note that cos()>sin() for all :>0), the circle S z =[` # C : `z = z sin()]/1 2:, ;. B the Cauch integral formula, for such z, as desired. K } d dz (z) } = } 1 (`) 2?i d`} Sz (`z) 2 1 max (`) ` # S z z sin() max ` # S z =max ` # S z } (`) ` } (`) ` max ` } ` # S z } 1+sin() sin() 1 z sin() =o(1) ( z, z # 1 : ), We are now read to prove our main result. Proof of Theorem 4.3 [(i) O (ii)]. Assume that + # D[# 0, 1 ]. The Nevanlinna representation (2.2) of F + (z) ields for >0 IF + (i)=+ 1 L _() t 2 d_(t) = + 1 L _( )+ 1 _(0) + 1 o(1) ().

19 FREE ADDITIVE CONVOLUTION 377 In virtue of Proposition 3.3 and Proposition 3.6, we onl need to show that either L _ # R 0 or L _ =0. B Lemma 5.2 there exists l # R 0 such that IF + (i)=+ 1 l()+1 l() o(1) (). Hence L _ ( )+_ (0) tl() (), (5.5) which proves the assertion. [(ii) O (i)]. Suppose that H + # R 0 (and so b Proposition 3.6 L + # R 0 ). Without loss of generalit we can assume that + has mean zero. Combining (3.7) and (3.8) in Proposition 3.6, we get F + (i)=i 1 i H +( )+ 1 }(), with }( )=o(h + ()) as. Fix :>0. B Lemma 5.3, ddzf 1 + (z)= 1+=(z), with =(z)=o(1) as z, z # 1 :, and since it is clear from the above equation that F + (i)#1 :, for large enough, we have that i=f 1 + (F +(i)) =F 1 + \ i 1 i H +( )+ 1 }() + =F 1 + (i)1 i H +( )+ 1 }()+ # =(!) d!, where # is the segment joining the points i and i(i) 1 H + ( )+ 1 }( ). Notice that =(!) d! } }1 # (H +( )+ }() ) sup =(!) = 1! # # H +() o(1) (), which implies, + (i)= 1 i H +( )+ 1 H +() o(1) (). (5.6) Since H + # R 0, b (P2) it is possible to find a positive sequence [B n ] n=1 such that n B 2 n H + (B n )t1 (n), (5.7)

20 378 VITTORINO PATA thus defining n =D 1Bn + }}}D 1Bn +, n times we have that, for ever n, the functions, n, are defined in a truncated cone 1 :, ;, and for >;,, n (i)= n B n, + (ib n ). Recalling that B n as n, and using (5.6), the above equation becomes, n (i)= 1 n H + (B n )(1+o(1)) i as or n. Therefore, in virtue of (5.7), we get B 2 n B (P1), for and n large enough, lim, n (i)= 1 n i. (5.8) H + (B n ) H + (B n ) =c(b n ) exp \ c( ) B n B n and =(t) <1 for tb n. Then we get =(t) dt + t, exp \ B n B n =(t) dt t } B n +exp B n =(t) dt t B n }exp 1 B n t dt=, and we conclude that, for such n, H + (B n ) H + (B n ) (1+o(1)) (). B (5.7), we also know that for n large enough, nb 2 n H + (B n ) is bounded, thus }, n (i) } = 1 2\ n +(B n ) H B 2 + (B n ) +\H (1+o(1))0 () (5.9) n H + (B n ) + uniforml as n. In force of (5.8) and (5.9), we can appl Proposition 2.1 (iii), and thus n converges weakl to a measure, and, (i)=(i) 1 for ever >;. The identit theorem then implies that, (z)=z 1 for ever z # 1 :, ;, which in turn implies that =# 0, 1. K

21 FREE ADDITIVE CONVOLUTION 379 Implication (ii) O (i) of the next theorem is in [9]. However, in view of Theorem 4.3 we can provide a three-line proof. Proof of Theorem 4.4 [(ii) O (ii)]. If + # ND[# 0, 1 ], the function l in Lemma 5.2 can be chosen to be l( )=b 2 (indeed, for such an l, (5.2) holds). Thus b (5.5), the measure _ occurring in the Nevanlinna representation (2.2) of F + has finite variance, and _ (0) +_ (2) =b 2. Hence b Corollar has finite variance b 2. [(ii) O (i)]. Suppose that + has finite variance b 2 and mean zero. Then (5.6) becomes, + (i)= b2 i +1 o(1) (), and the rest of the proof is exactl like in Theorem 4.3, with B n =- nb. K ACKNOWLEDGMENTS I thank Hari Bercovici for his valuable help in the realization of this paper. I also profited from conversations with Richard Bradle. REFERENCES 1. H. Bercovici and V. Pata, The law of large numbers for free identicall distributed random variables, Ann. Probab., to appear. 2. H. Bercovici and D. Voiculescu, Le vhinc$ in tpe theorems for multiplicative and additive free convolution, Pacific J. Math. 153 (1992), H. Bercovici and D. Voiculescu, Free convolution of measures with unbounded support, Indiana U. Math. J. 42 (1993), H. Bercovici and D. Voiculescu, Superconvergence to the central limit theorem and failure of the Crame r theorem for free random variables, Probab. Theor Relat. Fields, to appear. 5. N. H. Bingham, C. M. Goldie, and J. L. Teugels,``Regular Variation,'' Wile, New York, W. Feller, ``An Introduction to Probabilit Theor and Its Applications,'' Cambridge Univ. Press, Cambridge, UK, I. A. Ibragimov and Yu. V. Linnik, ``Independent and Stationar Sequences of Random Variables,'' Wolters-Noordhoff, Groningen, J. M. Lindsa and V. Pata, Some weak laws of large numbers in non-commutative probabilit, Math. Z., to appear. 9. H. Maassen, Addition of freel independent random variables, J. Funct. Anal. 106 (1992), V. Pata, Le v tpe characterization of stable laws for free random variables, Trans. Amer. Math. Soc. 347 (1995),

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