NEW EXAMPLES OF CONVOLUTIONS AND NON-COMMUTATIVE CENTRAL LIMIT THEOREMS

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1 QUANTUM PROBABILITY BANACH CENTER PUBLICATIONS, VOLUME 43 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 998 NEW EXAMPLES OF CONVOLUTIONS AND NON-COMMUTATIVE CENTRAL LIMIT THEOREMS MAREK BOŻEJKO andjanusz WYSOCZAŃSKI Insiue of Mahemaics, Universiy of Wroc law pl. Grunwaldzki 2/4, Wroc law, Poland Absrac. A family of ransformaions on he se of all probabiliy measures on he real line is inroduced, which makes i possible o define new examples of convoluions. The associaed cenral limi heorems are sudied, and examples of he limi measures, relaed o he classical, free and boolean convoluions, are shown.. Inroducion. By sudying series of examples of known limi heorems we came up wih an idea of new convoluions of measures. We found a family U, 0, of coninuous ransformaions, acing on probabiliy measures on he real line, which allowed us o define he convoluions and sudy associaed cenral limi heorems. The mos insrucive way of describing he ransformaions is given by observing heir acion on he Cauchyransformofameasure,givenin he formofaconinuedfracionbyaheorem of Sieljes. The crucial ideological role in our consrucion is played by relaions beween momens and cumulans of a given measure; hough he consrucion iself does no require ha he measure has finie momens. 2. General form of he non-commuaive cenral limi heorem and examples of limi measures. In [BSp] he following general form of he non-commuaive cenral limi heorem was obained: Theorem. Le B be a unial -algebra wih a sae ϕ and le b i = b i, i =,2,3,... be a sequence of self-adjoin elemens in B, saisfying he following assumpions:. for all posiive inegers n and all sequences i(),i(2),...,i(n) of indices, ϕ(b i() b i(2)...b i(n) ) = 0 whenever here exiss an index i(k), k n, which is disinc from all oher indices; 99 Mahemaics Subjec Classificaion: Primary 46L50; Secondary 60F05. Research parially suppored by KBN gran 2P03A0508. Thepaperisinfinalformandnoversionofiwillbepublishedelsewhere. [95]

2 96 M. BOŻEJKO AND J. WYSOCZAŃSKI 2. he expression ϕ(b i() b i(2)...b i(n) ) is invarian under all permuaions of he indices: for every permuaion π of he se N of posiive inegers ϕ(b i() b i(2)...b i(n) ) = ϕ(b π(i()) b π(i(2))...b π(i(n)) ). Moreover, for each n N le S n = n (b +b b n ); hen for each power k here exiss he limi lim n ϕ(sk n) = { 0, if k is odd, π P 2(,2,...k) (π), if k is even, () where P 2 (,2,...,2r) is he se of all pair pariions of he se {,2,...,2r}, and he funcion is given by he value (π) = ϕ(b i() b i(2)...b i(2r) ) common for all hese sequences i(),i(2),...i(2r), which saisfy he condiion: i(s) = i(m) if and only if s and m belong o he same block of he pariion π. For special choices of funcions one can obain, in he limi, momens of various known measures. This inspired he quesion of wha measures may appear as he limi in a non-commuaive cenral limi heorem. A conribuion o his problem is conained in he nex secions. Examples.. For (π) he limi is he gaussian measure 2π e x2 2 dx, as he limi momens are m µ (2n) = 3... (2n ) = (2n )!!. 2. For (π) = ( ) i(π) one ges he fermionic case wih he limi measure 2 (δ +δ ); here i(π) is he number of inversions of a pariion π. 3. For {, if π has no inversions, (π) = 0, if π has inversions, one ges he free case, i.e. he semi-circular Wigner measure 2π χ [ 2,2](x) 4 x 2 dx wih (even) momens being he Caalan numbers 2n n+( n). 4. For q (π) = q i(π), where q, he resuling q-gaussian measure has he Jacobi θ funcion as is densiy, see [BSp2]. 5. In he case of r (π) = r n b(π) where b(π) is he number ofblocks of π and 0 r he limi measures were obained in [BSp]. For r = N he measure µ r is he free produc of N copies of a dilaion of he gaussian measure. 6. For negaive r wih r 0, by a sligh modificaion of he previous case, namely by puing r (π) = ( r) n b(π) ( ) i(π) one can also compue he limi measure using he resuls of [BSp], Theorem 7. The even momens of he limi measure µ r are given by he formula m µr (2n) = (+r) inn(π) (2) π NC 2(2n) where inn(π) is he number of inner blocks in a pariion π. In paricular for r = N he limi measure µ r is he N-fold free convoluion of a dilaion of he wo-poin measure µ = 2 (δ +δ ). Anoher series of examples come from he group case. The general scheme in hese cases is he following. Given a (discree) group G and an infinie se S of is generaors

3 CONVOLUTIONS AND CLT 97 consider a random walk associaed wih he Cayley s graph of he pair (G,S). In oher words, define S n = n (λ(s j )+λ(s j )) 2n j= where λ(s) is he lef ranslaion operaor by s, and le ϕ(y) =< Yδ e,δ e > be he sandard sae on he algebra of all bounded operaors on l 2 (G). We sudy he limi lim n ϕ(sn), k which is of he form esablished above, wih b j = 2 (λ(s j ) + λ(s j )), provided he assumpions () and (2) are saisfied. Examples 2..LeG = F be afreegrouponinfiniely manygeneraorss = {s j : j =,2,3,...}; hen one ges {, if π has no inversions, (π) = 0, if π has inversions, and S n ends -weakly o he Wigner semi-circular disribuion. 2. If G is a free abelian (Coxeer) group wih he se of generaors S = {s j : j =,2,3,...} which saisfy he relaions s j s k = s k s j and s k = s k ; hen (π) and S n ends o he gaussian measure. 3. Le G be a free nilpoen group of class 2 or more; hen {, if π has no inversions, (π) = 0, oherwise, and S n = 2n n j= (λ(s j)+λ(s j )) ends o he Wigner semi-circular disribuion. 3. -ransformaion and -convoluion. For a given probabiliy measure µ wih compac suppor on he real line R, is Cauchy ransform G µ is defined for z C + = {z C : Iz > 0}, by: G µ (z) = + dµ(x) x and, by a heorem of Sieljes (see [AkG]), can be expressed as a coninued fracion: G µ (z) = (3) a b a 2 b 2 where he numbers a,a 2,... disappear if he measure is symmeric, and hey, ogeher wih he numbers b,b 2,b 3,... come from a recurrence formula for he polynomials orhogonal wih respec o he measure µ. The Cauchy ransform of a measure µ can be also expressed by he momens m µ (k) = + x k dµ(x)

4 98 M. BOŻEJKO AND J. WYSOCZAŃSKI of he measure as he series G µ (z) = m µ (k)k k=0 The Voiculescu s R-ransform r µ of he measure is hen defined by he equaliy (G µ ) (z) = +r µ (z) which involves he inverse of he Cauchy ransform. The R-ransform is nohing else bu he generaing funcion of he sequence of free cumulans (r µ (k)) k=0 of he measure. The free (addiive) convoluion of a pair of measures µ and µ 2 is defined, following Voiculescu, o be he measure µ for which r µ (k) = r µ (k)+r µ2 (k). The relaion beween he sequence of momens (m µ (k)) and he sequence of free cumulans (r µ (k)), found by Speicher (see [Sp]), is he following: m µ (n) = s,...,s k s +...+s k =n k r µ (k) m µ (s )... m µ (s k ) (4) Remark. Our noaion of n-h momen m µ (n) of a measure µ and of n-h cumulan r µ (n) of a measure µ slighly differs from ha used elsewhere, which is m n (µ) for n-h momen and r n (µ) form n-h cumulan. Nowwedefinehe-ransform.Lebeaposiiverealnumberandleµbeacompacly suppored measure on he real line. Then he funcion G µ (z) defined for z C + by he formula: G µ (z) = +( )z (5) G µ (z) urns ou o be he Cauchy ransform of a probabiliy measure denoed by µ. This is a consequence of he following heorem, ha can be found, for example, in ([Ma]): Theorem 2 (Nevanlinna). A funcion F(z) is he reciprocal of he Cauchy ransform of a probabiliy measure on he real line if and only if here exiss a posiive measure ρ and a real number a such ha for Iz > 0 F(z) = a+z + + +xz x z dρ(x). Corollary3. For a pair of probabiliy measures ρ and ν on he real line, and a real number 0 here exiss a probabiliy measure µ such ha G µ (z) = G ν (z) + ( ) G ρ (z) where z C +. This follows direcly from he Nevanlinna s heorem. For our special choice of he measure ρ = δ 0, we ge a lile more:

5 CONVOLUTIONS AND CLT 99 Corollary4. For a given probabiliy measure µ on he real line and a non-negaive number 0, here exiss a (unique) probabiliy measure µ such ha G µ (z) = G µ (z) +( )z where z C +. Proof. I follows from he Nevanlinna s heorem ha G µ (z) +( )z = F(z)+( )z = a+z + + +xz x z d(ρ)(x) is he reciprocal of he Cauchy ransform of he probabiliy measure denoed by µ. Definiion. The measure µ is called he -ransform of a measure µ and he ransformaion U : µ µ is called he -ransformaion. The following properies of -ransformaion are direc consequences of he definiion: Proposiion5. For a probabiliy measure µ and real numbers,s 0, he following properies are saisfied:. (U ) 0 is a muliplicaive semigroup: U s (U (µ)) = U s (µ); 2. dilaions of a measure commue wih U : D λ (U (µ)) = U (D λ (µ)); 3. U (µ) µ in he -weak opology, if ; 4. U and U are inverses of each oher; 5. U (δ a ) = δ a for any real number a; 6. The -ransformaion is coninuous in he -weak opology of measures: if µ n µ hen U µ n U µ. Therefore, he mapping U is a muliplicaive -weakly coninuous ransformaion, which commues wih dilaions of measures. The -ransform of a measure is he inverse of he /-ransform of he measure. I is insrucive o idenify he acion of -ransformaion on he Cauchy ransformaion of a measure, given in he form of a coninued fracion: G µ (z) = (6) b a a 2 b 2 a 3 b 3 so he acion looks quie simple: only he firs level (i.e. a,b ) is muliplied by. The -convoluion is defined in he following way. Given wo probabiliy measures µ and ν on he real line, a non-negaive number, and a convoluion (for which he classical convoluion, free Voiculescu convoluions, boolean free convoluion, and oher convoluions may serve) one defines: µ ν = (µ ν ) / = U (U (µ) U (ν)) (7)

6 00 M. BOŻEJKO AND J. WYSOCZAŃSKI The -convoluion provides a large new class of convoluions, which can be sudied from he poin of view of non-commuaive cenral limi heorem. In he case of he classical muliplicaive convoluion (i.e. on he muliplicaive group of posiive numbers) of wo poin-mass measures δ a and δ b, wih a,b posiive, heir -(classical) convoluion is no δ ab bu δ ab. On he oher hand he -(free) boolean convoluion commues wih -ransform, which seems o be an excepional case. In [BLS] he concep of c-free (i.e. condiionally free) convoluion was developed, in which c-free cumulans played crucial role. Le us recall ha if r ν (z) is he free cumulan (generaing) funcion of a measure ν, hen i is relaed o he Cauchy ransform of ν by he formula: G ν (z) = r ν(g ν (z)). (8) Le us also recall ha he c-free convoluion is defined for pairs of measures (and more general, for pairs of saes on unial *-algebras), say (µ,ν ) and (µ 2,ν 2 ), where for he second erms one applies jus he free convoluion. The c-free cumulan (generaing) funcion T (µ,ν) (z) = n=0 T (µ,ν)(n)z n, which depends on he given pair of measures (µ,ν), is relaed o he Cauchy ransform of he firs measure µ by he formula: G µ (z) = T (µ,ν)(g ν (z)). (9) As he firs formula is equivalen o he formula relaing momens wih free cumulans, he second one is equivalen o he following relaion beween momens and c-free cumulans: m µ (n) = T (µ,ν) (k) m ν (s )... m ν (s k )m µ (s k ). (0) s,...,s k s +...+s k =n k We shall consider a special case of he above consrucion, namely when ν = µ. In his case he wo formulas above combined wih he definiion of he -ransform give he relaion: r µ (n) = T (µ,ν) (n). () Therefore one ges he following special case of he formula for c-free cumulans: m µ (n) = r µ ( B ) T (µ,ν) ( B ) = R (π) inn(π) π NC(n) B π B inner B π B ouer where B is he number of elemens in a block B, and R (π) = R (µ,ν) (π) = B πt (µ,ν) ( B ). π NC(n) This formula is an exension o all q of he formula obained for q 0 in [BSp] in he proof of Theorem 7, wih = + q, since in our case can be any non-negaive number (cf. Example.6). 4. Cenral limi heorems for -convoluions. As a preparaion o he formulaion of cenral limi heorems we sar wih he following

7 CONVOLUTIONS AND CLT 0 Proposiion 6. If a given convoluion is associaive, hen for any posiive number he associaed -convoluion is also associaive. Proof. By definiion, for arbirary measures µ,ν,ρ (µ ν) ρ = U (U (U (U µ U ν)) U ρ) = U ((U µ U ν) U ρ). From hese equaliies he resul easily follows. Le D λ µ be he dilaion of a measure µ by a number λ, defined as D λ µ(a) = µ(λ A) for an arbirary measurable se A. In cenral limi heorems one sudies he limi of a sequence of measures of he form D λ µ... D λ µ, which is he n-h -convoluionpower of he dilaion of a measure µ by an appropriae number λ. I is herefore essenial o know ha he expression makes sense, i.e. ha he convoluion is associaive. Our cenral limi heorem has he following form Theorem7. Le µ be an arbirary probabiliy measure on he real line, wih mean zero and wih second momen equal o. Le also be a non-negaive number and be a given convoluion. Then he sequence of measures D n µ... D n µ ends in he -weak opology o a measure ν (), which is a ransformaion of he cenral limi measure ν for he convoluion. Proof. For a fixed n le λ = n. Then he sequence of he n-fold -convoluion of he measure µ is of he form D λ µ... D λ µ = U (U D λ µ... U D λ µ) = U (D λµ... D λ µ ). Since he measure µ = U µ has he second momen, he sequence D λ µ... D λ µ has he -weak limi D (ν), where ν is he cenral limi measure for he -convoluion, wih m ν (2) =. Since he Cauchy ransform of ν is given by a coninued fracion of he form G ν (z) = b 2 we obain G D (ν)(z) = b 3 b 2 b 3

8 02 M. BOŻEJKO AND J. WYSOCZAŃSKI and herefore he applicaion of he ransformaion U o he measure D (ν) gives he measure ν (), which has he following Cauchy ransform: G ν ()(z) = G U (D (ν)) (z) =. (2) b 2 b 3 Hence he sequence under consideraion has he -weak limi U (D (ν)) = ν(), by he coninuiy of he -ransformaion, since m µ () = m µ () = 0 and m µ (2) = m µ (2) =. Remark. The above coninued fracions combined may serve as a definiion of he ransformaion ν ν (). Since he coefficiens b k are non-negaive, also has o be non-negaive in his seing. Examples 3. Our hree basic examples are he following.. If is he classical convoluion of measures, hen he limi measure is he ransformaion ν ν () of he gaussian measure dν(x) = 2π e x2 2 dx. As he gaussian measure has he Cauchy ransform G ν (z) = 2 3 he limi measure has he Cauchy ransform G ν ()(z) = 2 3. (3) 2. If is he free convoluion, hen he limi is he ransformaion ν ν () of he Wigner semi-circular disribuion dν(x) = 2π χ [ 2,2](x) 4 x 2 dx. As he Wigner measure has he Cauchy ransform G ν (z) =

9 CONVOLUTIONS AND CLT 03 he limi measure in his case has he Cauchy ransform G ν ()(z) =. (4) 3. If is he boolean convoluion, hen we have a lile more. In his case he ransformaion U commues wih he convoluion, so he limi measure 2 (δ +δ ) remains unchanged. The Cauchy ransform of his measure is G µ (z) =. (5) z References [AkG] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operaors in Hilber Space, Ungar, New York, 963. [BLS] M. Bożejko, M. Leiner and R. Speicher, Convoluion and limi heorems for condiionally free random variables, Pacific J. Mah. 75, No. 2 (996), [BSp] M. Bożejk o and R. S p eich er, Inerpolaion beween bosinic and fermionic relaions given by generalized Brownian moions, Mah. Z. 222 (996), [BSp2] M. Bożejko, B. Kümmerer and R. Speicher, q-gaussian Processes: Noncommuaive and Classical Aspecs, Comm. Mah. Phys. 85 (997), [Ma] H. Maassen, Addiion of Freely Independen Random Variables, J. Func. Anal. 06 No. 2 (992), [Sp] R. S p eich er, Muliplicaive funcions on he laice of non-crossing pariions and free convoluion, Mah. Ann. 298 (994),