Tree convolution for probability distributions with unbounded support
aa r X i v : . [ m a t h . OA ] F e b TREE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITHUNBOUNDED SUPPORT
ETHAN DAVIS, DAVID JEKEL, AND ZHICHAO WANG
Abstract.
We develop the complex-analytic viewpoint on the tree convolutions studied by thesecond author and Weihua Liu in [27], which generalize the free, boolean, monotone, and orthogonalconvolutions. In particular, for each rooted subtree T of the N -regular tree (with vertices labeled byalternating strings), we define the convolution ⊞ T ( µ , . . . , µ N ) for arbitrary probability measures µ , . . . , µ N on R using a certain fixed-point equation for the Cauchy transforms. The convolutionoperations respect the operad structure of the tree operad from [27]. We prove a general limittheorem for iterated T -free convolution similar to Bercovici and Pata’s results in the free case [12],and we deduce limit theorems for measures in the domain of attraction of each of the classical stablelaws. Introduction
In [45, 46], Voiculescu introduced free independence, which provided a probabilistic viewpoint onfree products of operator algebras. Two other forms of non-commutative independence were stud-ied in non-commutative probability theory around the year 2000: boolean independence in [43] andmonotone independence in [36, 37]. Besides classical independence, these are the only types of in-dependence that provide an associative natural product operation on non-commutative probabilityspaces [41, 11, 38, 39]. However, there are many other types of independence broadly defined. For in-stance, Lenczewski defined m -free independences intermediate between free and boolean independence[31]. One can combine several algebras using a mixture of classical and free independence [35, 44],boolean and monotone independence [47], or boolean and free independence [29]. The notions of c -free [18, 4] and c -monotone [24, 34] independence are another way of combining free or monotoneindependence with boolean independence, using pairs of states.Weihua Liu and the second author defined a general family of non-commutative independencesassociated to rooted trees whose vertices are labeled by alternating strings [27], which would serveas a general framework for studying various convolution operations and the relationships betweenthem, such as the relation between free, monotone, and subordination convolution in [32, 33]. Theindependences defined by trees include free, monotone, and boolean independence; m -free indepen-dence; mixtures of free, boolean, and monotone independence. The introduction of [27] noted threeviewpoints on non-commutative independence (1) operator models, (2) combinatorics of moments,and (3) complex analysis of Cauchy transforms, of which that paper focused on only the first two.Our present goal is to develop the complex-analytic viewpoint.To set the stage, let us recall some of the main ideas of [27]. Let T N, free be the tree whose vertices arealternating strings on the alphabet [ N ] = { , . . . , N } (strings where consecutive letters are distinct)and where two strings are adjacent precisely when one is obtained by appending one letter to the leftof the other. Let Tree( N ) be the set of rooted subtrees of T N, free , where the root is the empty string.Each T ∈
Tree( N ) describes a way of combining N Hilbert spaces with unit vectors ( H , ξ ), . . . ,( H N , ξ N ) into a new Hilbert space ( H, ξ ) akin to the free product of pointed Hilbert spaces, which iscalled the T -free product of pointed Hilbert spaces [27, § T -freeconvolution: Suppose X j is a bounded operator on H j whose spectral measure with respect to ξ j is Mathematics Subject Classification.
Primary: 46L53, Secondary: 46L54, 05C76, 60F05, 60E07.Jekel was supported by NSF grant DMS-2002826. The data for the figures was generated using Sage on Cocalc, andthe pictures were created with TikZ.. µ j . If ˜ X , . . . , ˜ X N are the corresponding operators on the product space ( H, ξ ), then the convolution ⊞ T ( µ , . . . , µ N ) is the spectral measure of ˜ X + · · · + ˜ X N with respect to ξ . (In fact, all of this wasdone in [27] in the more general setting where Hilbert spaces are replaced by B - B -correspondencesfor some C ∗ -algebra B , and µ j is a B -valued law. But at present we are only concerned with the case B = C where the objects reduce to Hilbert spaces and compactly supported probability measures on R .)In order to relate various convolution operations, the family (Tree( N )) N ∈ N was equipped with thestructure of a topological symmetric operad, and the convolution operations were shown to respectthis operad structure [27, § T ∈
Tree( k ) and T ∈ Tree( n ), . . . , T k ∈ Tree( n k ),there is a well-defined composition T ( T , . . . , T k ) ∈ Tree( n + · · · + n j ) which satisfies ⊞ T ( T ,..., T k ) ( µ , , . . . , µ ,n , . . . . . . , µ k, , . . . , µ k,n k )= ⊞ T ( ⊞ T ( µ , , . . . , µ ,n ) , . . . , ⊞ T k ( µ k, , . . . , µ k,n k ))where µ i,j is a compactly supported probability measure on R . Many known convolution identitiescan be proved in this framework [27, § T -free convolution into booleanand orthogonal convolutions, which generalizes the decompositions of additive free convolution in [32].Let br j ( T ) = { s ∈ T N, free : js ∈ T } , where js denotes the string obtained by appending j to the startof the string s . Let ⊎ denote the boolean convolution and ⊢ the orthogonal convolution (see Examples4.7 and 4.8 below). Then(1.1) ⊞ T ( µ , . . . , µ N ) = ] j ∈ [ N ] ∩T [ µ j ⊢ ⊞ br j ( T ) ( µ , . . . , µ N )]for compactly supported probability measures on R . This relation is convenient for the complex-analytic viewpoint because the boolean and orthogonal convolutions have simple expressions in termsof the K -transform (an analytic function related to the Cauchy transform).In this paper, we will use (1.1) to define the T -free convolution for arbitrary probability measureson R . More precisely, in Theorem 4.1, we will show that there is a unique family of operations ⊞ T on probability measures that satisfies (1.1) and depends continuously on T (with respect tolocal convergence with respect to the root vertex). The convolution ⊞ T ( µ , . . . , µ N ) also dependscontinuously on µ , . . . , µ N and agrees in the compactly supported case with the prior definitionfrom [27]. Because (1.1) so directly relates with the K -transforms of measures, we can give self-contained proofs of the basic properties of T -free convolution without relying on operator models oron approximation of general probability measures with compactly supported ones, making the proofsin this paper essentially independent from [27]. In particular, in §
5, we show directly from Theorem4.1 that the convolution operation on arbitrary measures respects the operad structure just as in thecompactly supported case.In § §
7, we discuss limit theorems for T -free independence. Often when a new type of additiveconvolution is introduced, a central limit theorem and Poisson limit theorem are proved in the samepaper or soon thereafter, as in e.g. [45, 18, 43, 20, 37, 47, 29, 27]. In classical probability, more generallimit theorems for additive convolution are closely related to the study of infinitely divisible and stabledistributions, as well as the L´evy-Khintchine formula that classifies infinitely divisible distributions µ in terms of some other measure σ and real number γ ; see [21]. Similar results have been obtainedfor non-commutative independences, both in the scalar-valued and the operator-valued settings; seefor the free case [46, 13, 16, 42, 12, 40, 1], for the boolean case [43, 40, 1], for the monotone case[37, 6, 22, 23, 25, 2, 3, 26], for the c -free case [28, 10]. One of the most influential works on the topicwas Bercovici and Pata’s paper [12]. They showed that if µ ℓ is a sequence of measures and n ℓ is asequence of natural numbers tending to ∞ , then µ ∗ k ℓ ℓ converges to a measure ν ∗ if and only if µ ⊞ k ℓ ℓ converges to a measure ν ⊞ if and only if µ ⊎ k ℓ ℓ converges to a measure ν ⊎ , and the correspondencebetween ν ∗ , ν ⊞ , and ν ⊎ is described in the terms of the respective L´evy-Khintchine formulas. From thisgeneral statement, they deduced free and boolean analogs of all classical limit theorems for additive REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 3 convolution, and in particular limit theorems for the domains of attraction corresponding to eachclassical stable distribution.For a general choice of a tree
T ∈
Tree( N ), it is unclear how to define the k th convolution powerfor arbitrary k , as discussed in [27, § k -fold composition of T with itself,denoted T ◦ k ; the corresponding convolution is an N k -ary operation. Let n ( T ) denote the numberof neighbors of the root vertex. When n ( T ) ≥
0, [27, §
9] classified infinitely divisible laws in the B -valued setting under certain boundedness assumptions. In this paper, in Theorem 6.1, we obtain ananalogue of one direction of Bercovici and Pata’s main result for arbitrary probability measures on R .If µ ⊎ n ( T ) kℓ ℓ → ν , then ⊞ T ◦ kℓ ( µ ℓ , . . . , µ ℓ ) converges to a measure BP ( T , ν ) (Theorem 6.1). We do notknow whether the converse implication holds. Nonetheless, the theorem already contains the “morepractical” implication, where the hypothesis is the relatively easy-to-check condition about booleanconvolution and the conclusion describes convergence for general trees T (and in fact gives a uniformrate over convergence over all T ∈
Tree( N )). In particular, Theorem 6.1 allows us to deduce limittheorems corresponding to each of the classical domains of attraction in § § T -free convolutions and limit theoremsin § §
2, we explain background material on probability measureson R on their Cauchy transforms. In §
3, we review the operad of rooted trees from [27] and establishmore of its basic properties. In §
4, we define the T -free convolution of arbitrary probability measureson R . In §
5, we show that the convolution operations respect the operad structure. In §
6, we provethe general limit theorem. In §
7, we deduce as special cases limit theorems for each of the domains ofattraction from classical probability theory. In §
8, we propose questions for future research.2.
Cauchy transforms of probability measures M ( R ) denotes the space of finite positive Borel measures on R , P ( R ) denotes the space of proba-bility measures, equipped with the weak topology (that is, the weak- ∗ topology when viewed insidethe dual of C ( R )). Recall that P ( R ) is metrizable using the L´evy distance d L ( µ, ν ) := inf n ǫ > µ (( −∞ , x − ǫ )) − ǫ ≤ ν (( −∞ , x )) ≤ µ ( −∞ , x + ǫ )) + ǫ ) for all x ∈ R o . Definition 2.1.
For a finite measure µ on R , the Cauchy-Stieltjes transform is given by G µ ( z ) = Z R z − t dµ ( t ) . The F -transform is given by F µ ( z ) = 1 /G µ ( z ) , and we also define K µ ( z ) = z − F µ ( z ) . Let Hol( H , − H ) be the space of holomorphic functions H → − H . Then Hol( H , − H ) is a normalfamily if we view the target space as sitting inside the Riemann sphere, hence the topology of pointwiseconvergence on Hol( H , − H ) agrees with the (metrizable) topology of local uniform convergence. Lemma 2.2.
For each m > , the map { µ ∈ M ( R ) : k µ k ≤ m } → Hol( H , − H ) : µ G µ is a homeomorphism onto its image, where we use the weak- ∗ topology on M ( R ) and the topology oflocal uniform convergence on Hol( H , − H ) . Next, we recall the famous theorem of Nevanlinna that characterizes Cauchy transforms of proba-bility measures as functions G ( z ) such that zG ( z ) → z → ∞ non-tangentially in H . To state thetheorem, recall the following definitions of cones and non-tangential convergence.For a >
0, let Γ a ⊆ H be the cone Γ a := { z : Im z ≥ a | z |} . ETHAN DAVIS, DAVID JEKEL, AND ZHICHAO WANG
We also define for a, b, c >
0, the regionsΓ a,b := { z : Im z ≥ max( a | z | , b ) } . Definition 2.3.
Let Y be a topological space, and let F : H → Y . We say that F ( z ) → L as z → ∞ non-tangentially if for every a >
0, lim z →∞ z ∈ Γ a F ( z ) = y, or equivalently, for every a > U of y , there exists b > F ( z ) ∈ U for every z ∈ Γ a,b . Theorem 2.4 (Nevanlinna) . Let G : H → C and m > . The following are equivalent:(1) G is the Cauchy transform of some measure of total mass m .(2) G maps H into − H and zG ( z ) → m non-tangentially as z → ∞ .(3) G maps H into − H and lim y →∞ iyG ( iy ) = m over y > . Corollary 2.5.
A function F is the F -transform of some probability measure on R if and only if F maps H into H and F ( z ) /z → as z → ∞ non-tangentially. Similarly, K is the K -transformof some probability measure on R if and only if K maps H into − H and K ( z ) /z → as z → ∞ non-tangentially.Proof. The first claim is immediate from the theorem since F µ ( z ) = 1 /G µ ( z ). Similarly, for the secondclaim, the only thing that remains to prove is that Im K µ ( z ) ≤ µ . For c >
0, observe that the region Ω c = { x + iy : Im(1 / ( x + iy )) ≥ c } = { x + iy : − y/ ( x + y ) ≥ c } = { x + iy : c ( x + y ) + y ≤ } is a disk and in particular is convex. For z ∈ H and t ∈ R , we have 1 / ( z − t ) ∈ Ω Im z , and hence G µ ( z ) = R R ( z − t ) − dµ ( t ) ∈ Ω Im z . Thus, Im F µ ( z ) ≥ Im z , or equivalently Im K µ ( z ) ≤ (cid:3) The following result is contained in [12, Proof of Proposition 2.6] and thus we leave the reader tolook up or reconstruct the proof.
Lemma 2.6. If Y is a tight family of probability measures, then zG µ ( z ) → as z → ∞ non-tangentially, uniformly over µ ∈ Y . Similarly, we have F µ ( z ) /z → and K µ ( z ) /z → as z → ∞ non-tangentially, uniformly for µ ∈ Y . An operad of rooted trees
Definition 3.1.
For N ∈ N , let [ N ] = { , . . . , N } . A string on the alphabet [ N ] is a finite sequence j . . . j ℓ with j i ∈ [ N ]. We denote by the i th letter of a string s by s ( i ). Given two strings s and s ,we denote their concatenation by s s . Definition 3.2.
A string j . . . j ℓ is called alternating if j i = j i +1 for every i ∈ { , . . . , ℓ − } . Definition 3.3.
Let T N, free be the (simple) graph whose vertices are the alternating strings on thealphabet [ N ] and where the edges are given by s ∼ js for every letter j and every string s that doesnot begin with j . Note that T N, free is an infinite N -regular tree. We denote the empty string by ∅ ,and we view ∅ as the preferred root vertex of the graph T N, free . Definition 3.4.
We denote by Tree( N ) the set of rooted subtrees of T N, free (that is, connectedsubgraphs containing the vertex ∅ ). Note that if T ∈
Tree( N ), then the edge set is uniquely determinedby the vertex set and vice versa. Thus, we may treat T merely as a set of vertices when it is notationallyconvenient. If s ∈ T and js ∈ T for some string s and some j ∈ [ N ], then we say that js is a child of s and s is the parent of js . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 5
Observation 3.5.
For a rooted tree
T ⊆ T N, free and ℓ ≥ , let B ℓ ( T ) ⊆ T N, free be set of stringsin T of length ≤ ℓ (or equivalently the closed ball of radius ℓ in the graph metric). Define ρ N : T N, free × T N, free → R by ρ N ( T , T ′ ) = exp( − sup { ℓ ≥ B ℓ ( T ) = B ℓ ( T ′ ) } ) . Then ρ N defines a metric on Tree( N ) (and in fact an ultrametric), which makes Tree( N ) into acompact metric space. As explained in [27, § k )) k ∈ N form a topological symmetric operad . (Forgeneral background on operads, see e.g. [30], and the complete definition is also explained in [27].)We have already described the topology. The operad structure consists of composition mapsTree( k ) × Tree( n ) × · · · × Tree( n k ) → Tree( n + · · · + n k ) : ( T , T , . . . , T k )
7→ T ( T , . . . , T )for each k ∈ N and n , . . . , n k ∈ N , which are given as follows. Let T ∈
Tree( k ) and T ∈ Tree( n ),. . . , T k ∈ Tree( n k ). Let N j = n + · · · + n j and N = N k . Define ι j : [ n j ] → [ N ] by ι j ( i ) = N j − + i , sothat [ N ] = F kj =1 ι j ([ n j ]). For a string s ∈ T n j , free , let ( ι j ) ∗ ( s ) denote the string obtained by applying ι j to each letter of s . Then we define T ( T , . . . , T k ) ∈ Tree( N ) to be the rooted subtree with vertexset(3.1) [ ℓ ≥ [ i ...i ℓ ∈T [ s j ∈T ij \{∅} for j ∈ [ ℓ ] ( ι i ) ∗ ( s ) . . . ( ι i ℓ ) ∗ ( s ℓ ) . In other words, the strings in T ( T , . . . , T k ) are obtained by taking a string t = i . . . i ℓ in T andreplacing each letter i j by a string s j from T i j , with the indices appropriately shifted by ι j : [ n j ] → [ N ].This composition operation satisfies the operad associativity axioms. It is also jointly continuous, andin fact, we have ρ N ( T ( T , . . . , T k ) , T ′ ( T ′ , . . . , T ′ k )) ≤ max( ρ k ( T , T ′ ) , ρ n ( T , T ′ ) , . . . , ρ n k ( T k , T ′ k )) , where T , T ′ ∈ Tree( k ) and T j , T ′ j ∈ Tree( n j ) for j = 1, . . . , k . This is because every string of length L the composition has the form ( ι i ) ∗ ( s ) . . . ( ι i ℓ ) ∗ ( s ℓ ) as above, where ℓ ≤ L and s , . . . , s ℓ havelength ≤ L .Finally, Tree is a symmetric operad, which means that there is a right action of the permutationgroup Perm( N ) on Tree( N ) that satisfies natural compatibility properties with the operad composition(see [30]). The permutation action on Tree( N ) is defined as follows: For a string s = j . . . j ℓ , let σ ( s ) = σ ( j ) . . . σ ( j ℓ ). Then for a tree T ⊆ T N, free , let T σ = { σ − ( s ) : s ∈ T } . This permutationaction is continuous (and in fact isometric) on Tree( N ).Central to this paper is the iterative formula [27, Proposition 6.8] which expresses convolutionsover a tree T in terms of the convolutions over the branches of T for each neighbor of the root vertex.To set the stage, we define the branch operations and describe how they interact with the topologicalsymmetric operad structure of Tree. Definition 3.6.
For j ∈ [ N ], we define br j : Tree( N ) → Tree( N ) ∪ { ∅ } by(3.2) br j ( T ) = { s ∈ T N, free : sj ∈ T } . This gives the branch of T rooted at the vertex j if j ∈ T and ∅ otherwise. Observation 3.7.
The map br j is a continuous (and in fact e − -Lipschitz) function from the clopenset {T ∈ Tree( N ) : j ∈ T } into Tree( N ) . Observation 3.8.
For
T ∈
Tree( N ) and σ ∈ Perm( N ) , we have br j ( T σ ) = br σ ( j ) ( T ) σ . In order to describe the relationship between the branch operation and operad composition, we needsome auxiliary notions. Let ψ : [ N ] → [ N ′ ]. For a string s = j . . . j ℓ on [ N ], let ψ ∗ ( s ) = ψ ( j ) . . . ψ ( j ℓ ).Viewing a tree T ∈
Tree( N ) as a set of strings, we may compute the image ψ ∗ ( T ) under the map ψ ∗ .Of course, if s is alternating, then ψ ∗ ( s ) is not necessarily alternating. Thus, ψ ∗ ( T ) will be an elementof Tree( N ′ ) if and only if ψ ∗ ( s ) is alternating for every s ∈ T , or in other words, ψ ∗ ( T ) ⊆ T N ′ , free . ETHAN DAVIS, DAVID JEKEL, AND ZHICHAO WANG
Lemma 3.9.
Let T , mono = {∅ , , , } . Let T ∈
Tree( k ) and T ∈ Tree( n ) , . . . , T k ∈ Tree( n k ) . Let N = n + · · · + n k and let ι j : [ n j ] → [ N ] be the inclusions as above. Let ψ : [ n j + N ] → [ N ] map thefirst n j points monotonically onto ι j ([ n j ]) and map the last N points monotonically onto [ N ] . Thenfor j ∈ [ k ] and i ∈ [ n j ] , we have (3.3) br ι j ( i ) ( T ( T , . . . , T k )) = ψ ∗ [ T , mono (br i ( T j ) , br j ( T )( T , . . . , T k ))] . Proof.
By the definition of operad composition (3.1), we can directly getbr ι j ( i ) ( T ( T , . . . , T k ))= [ si ∈T j ( ι j ) ∗ ( s ) [ [ ℓ ≥ [ i ...i ℓ j ∈T [ s t ∈T it \{∅} for t ∈ [ ℓ ] ( ι i ) ∗ ( s ) . . . ( ι i ℓ ) ∗ ( s ℓ ) [ [ ℓ ≥ [ i ...i ℓ j ∈T [ s t ∈T it \{∅} for t ∈ [ ℓ ] si ∈T j ,s = ∅ ( ι i ) ∗ ( s ) . . . ( ι i ℓ ) ∗ ( s ℓ )( ι j ) ∗ ( s ) . The branch of T ( T , . . . , T k ) rooted at the vertex ι j ( i ) has two parts: one is purely generated by the T j ; the other is generated by the composition of T j and the other T i , for i = j . For second part, wecan further split it into two parts. Moreover, by Definition 3.2, T ′ := br i ( T j ) = { s ∈ T n j , free : si ∈T j } ∈ Tree( n j ), and T ′ := br j ( T )( T , . . . , T k ) = [ ℓ ≥ [ i ...i ℓ j ∈T [ s t ∈T it \{∅} for t ∈ [ ℓ ] ( ι i ) ∗ ( s ) . . . ( ι i ℓ ) ∗ ( s ℓ ) ∈ Tree( N ) . Hence, by denoting ι ′ : [ n j ] → [ n j + N ] and ι ′ : [ N ] → [ n j + N ] by ι ′ ( i ) = i and ι ′ ( i ) = i + n j , wehave the following decomposition: T , mono ( T ′ , T ′ ) = [ s ∈T ′ ( ι ′ ) ∗ ( s ) [ [ s ∈T ′ \{∅} ( ι ′ ) ∗ ( s ) [ [ s i ∈T ′ i \{∅} , i =1 , ( ι ′ ) ∗ ( s )( ι ′ ) ∗ ( s ) . Notice that ψ ◦ ι ′ ( i ) = ι j ( i ), for i ∈ [ n j ] and ψ ◦ ι ′ ( i ) = i , for i ∈ [ N ]. Thus, ψ ∗ ◦ ( ι ′ ) ∗ ( s ) = ( ι j ) ∗ ( s ),for si ∈ T j , and ψ ∗ ◦ ( ι ′ ) ∗ ( s ) = s , for s ∈ T ′ . So, by applying ψ ∗ to T , mono ( T ′ , T ′ ), we can deduce(3.3). (cid:3) Next, we define isomorphism of rooted trees and describe how isomorphism relates to the branchmaps.
Definition 3.10.
Let T ∈ Tree( N ) and T ∈ Tree( N ). We say a map φ : T → T is a homomor-phism if φ ( ∅ ) = ∅ and for each vertex s ∈ T and each child s ′ of s , φ ( s ′ ) is a child of φ ( s ). We saythat φ is an isomorphism if it is a bijective homomorphism, and in this case, we write T ∼ = T . Observation 3.11.
Let T ∈ Tree( N ) and T ∈ Tree( N ) and let φ : T → T be an isomorphism.Then φ defines a bijection [ N ] ∩ T → [ N ] ∩ T and we have br j ( T ) ∼ = br φ ( j ) ( T ) . Definition 3.12.
For
T ∈
Tree( N ), let us write n ( T ) = | [ N ] ∩ T | m ( T ) = max s ∈T \{∅} |{ j ∈ [ N ] : js ∈ T }| , that is, n ( T ) is the number of children of the root vertex and m ( T ) is the maximum number ofchildren of any other vertex of the tree. Observation 3.13.
The quantities n ( T ) and m ( T ) are isomorphism-invariant. If T ∈
Tree( N ) , then max( n ( T ) , m ( T )+ 1) ≤ N , and T is isomorphic to some T ′ ∈ Tree( N ′ ) with N ′ = max( n ( T ) , m ( T )+1) . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 7
Proof.
The first claim is immediate. By construction, for
T ∈
Tree( N ), the root has at most N childrenand the other vertices have at most N − N ′ = max( n ( T ) , m ( T ) + 1), anyisomorphism class of trees where the root as at most N ′ children and the other vertices have at most N ′ − T N ′ , free . (cid:3) The final set of notation and results relates to compositions of several copies of the same tree; theseremarks will be used in § Definition 3.14.
Given trees T ∈ Tree( N ) and T ∈ Tree( N ), let T ◦ T := T ( T , . . . , T | {z } N times ) . This operation is associative because of the operad associativity property for Tree. Thus, thefollowing definition also makes sense without parentheses.
Definition 3.15.
For
T ∈
Tree( N ), let T ◦ k be given by T ◦ k := T ◦ · · · ◦ T | {z } k times . Lemma 3.16.
For T ∈ Tree( N ) and T ∈ Tree( N ) , we have n ( T ◦ T ) = n ( T ) n ( T ) m ( T ◦ T ) = m ( T ) n ( T ) + m ( T ) . Moreover, (as in [27, Lemma 8.7] ) for
T ∈
Tree( N ) with n ( T ) > , we have n ( T ◦ k ) = n ( T ) k m ( T ◦ k ) = m ( T ) n ( T ) k − n ( T ) − . Proof.
For j ∈ [ N ], let ι j : [ N ] → [ N N ] be given by ι j ( i ) = ( j − N + i . The neighbors of ∅ in T ◦ T have the form ι j ( i ) where j is a neighbor of ∅ in T and i is a neighbor of ∅ in T , and hence n ( T ◦ T ) = n ( T ) n ( T ).Next, consider the children of some non-root vertex of T ◦ T . This vertex has the form s =( ι j ) ∗ ( s ) . . . ( ι j ℓ ) ∗ ( s ℓ ) where j . . . j ℓ ∈ T and s , . . . , s ℓ ∈ T \ {∅} . There are two ways to append aletter to the front of this string and remain in T ◦ T . First, we could append a letter i to the frontof s in T to obtain ( ι j ) ∗ ( is )( ι j ) ∗ ( s ) . . . ( ι j ℓ ) ∗ ( s ℓ ); there are at most m ( T ) possible ways to dothis. Second, we could append ι j ( i ) to s for some j such that jj . . . j ℓ ∈ T and some i ∈ [ N ] ∩ T ;there are at most m ( T ) n ( T ) possible ways to do this. Thus, the number of children of s in T ◦ T isat most m ( T ) n ( T ) + m ( T ). To show that this number of children is achieved in T ◦ T \ {∅} , picksome j . . . j ℓ ∈ T \ {∅} with m ( T ) children, pick s ∈ T \ {∅} with m ( T ) children, and pick s ,. . . , s ℓ ∈ T \ {∅} arbitrarily. Then s = ( ι j ) ∗ ( s ) . . . ( ι j ℓ ) ∗ ( s ℓ ) will have exactly m ( T ) n ( T ) + m ( T )children in T ◦ T by the foregoing argument.Clearly, n ( T ◦ k ) = n ( T ) k follows by induction on k . For the next formula, note that m ( T ◦ ( k +1) ) = m ( T ◦ T ◦ k ) = m ( T ) n ( T ◦ k ) + m ( T ◦ k ) = m ( T ) n ( T ) k + m ( T ◦ k ) . Hence, m ( T ◦ k ) = m ( T ) + k − X j =1 [ m ( T ◦ ( j +1) ) − m ( T ◦ j )] = k − X j =0 m ( T ) n ( T ) j = m ( T ) n ( T ) k − n ( T ) − . (cid:3) Lemma 3.17.
Let T ∈ Tree( N ) , T ∈ Tree( N ) , T ′ ∈ Tree( N ′ ) , T ′ ∈ Tree( N ′ ) . If T ∼ = T ′ and T ∼ = T ′ as rooted trees, then T ◦ T ∼ = T ′ ◦ T ′ . ETHAN DAVIS, DAVID JEKEL, AND ZHICHAO WANG
Proof.
Let φ : T → T ′ and φ : T → T ′ be isomorphisms. Let ι j : [ N ] → [ N N ] be given by ι j ( i ) = ( j − N + i , and define ι ′ j analogously for N ′ and N ′ instead of N and N . Then we define anisomorphism ψ : T ◦ T → T ′ ◦ T ′ as follows. Any vertex of T ◦ T has the form ( ι j ) ∗ ( s ) . . . ( ι j ℓ ) ∗ ( s ℓ ),where j . . . j ℓ ∈ T (here ℓ ≥
0) and s i ∈ T \ {∅} . Now φ ( j . . . j ℓ ) must be a string of the samelength, so suppose that φ ( j . . . j ℓ ) = j ′ . . . j ′ ℓ . Then we define ψ (( ι j ) ∗ ( s ) . . . ( ι j ℓ ) ∗ ( s ℓ )) = ( ι ′ j ′ ) ∗ ( φ ( s )) . . . ( ι ′ j ′ ℓ ) ∗ ( φ ( s ℓ )) . Since φ and φ are isomorphisms, the right-hand side will realize every possible string from T ′ ◦T ′ , andin fact will be a bijection. The only thing left to prove is that ψ preserves parent-child relationships,and this is done by examining the two cases of children as in the proof of the previous lemma. (cid:3) Tree convolutions
The main result of this section is the following theorem:
Theorem 4.1.
There exists a unique function
Tree( N ) × P ( R ) N → P ( R ) : ( T , µ , . . . , µ N ) ⊞ T ( µ , . . . , µ N ) that is continuous in T and satisfies (4.1) K ⊞ T ( µ ,...,µ N ) ( z ) = X j ∈T K µ j ( z − K ⊞ br j ( T ) ( µ ,...,µ N ) ( z )) . In fact, this map is jointly continuous
Tree( N ) × P ( R ) N → P ( R ) . The convolution will be constructed by iteration to a fixed point similar to the description of freeand subordination convolutions in [9]. One of the main ingredients in the proof is the Earle-Hamiltontheorem, which is a fixed-point theorem for holomorphic functions between Banach spaces.
Definition 4.2 (See [48, 49, 50]) . Let X and Y be Banach spaces and let Ω be an open subset of X .A function f : Ω → Y is holomorphic if(1) For each x ∈ X , there exists r > B ( x, r ) ⊆ Ω and f ( B ( x, r )) is bounded.(2) For each x, x ′ ∈ X and φ ∈ Y ′ , the function C → C mapping z to φ [ f ( x + zx ′ )] is holomorphic onthe region where it is defined. Theorem 4.3 (Earle-Hamilton [19]) . Let X be a Banach space and Ω an open subset of X . Supposethat F : Ω → Ω is holomorphic, F (Ω) is bounded, and d ( F (Ω) , Ω c ) > . Then F has a unique fixedpoint in Ω and for any x ∈ Ω , the iterates F ◦ n ( x ) converge to the fixed point as n → ∞ . We also use the following lemma about K -transforms and truncated cones. For a > < b < c ,we define Γ a,b,c := { z : Im z ≥ max( a | z | , b ) , | z | ≤ c } . Note that Γ a,b,c is convex.
Lemma 4.4.
Let Y ⊆ P ( R ) be compact, let N > , and suppose that < a < a < a , < b < b < b < c < c < c . Then for sufficiently large t , we have µ ∈ Y, z ∈ Γ a ,tb ,tc , w ∈ Γ a ,tb ,tc = ⇒ z − N K µ ( w ) ∈ Γ a ,tb ,tc . Proof.
Let ǫ ( t ) = sup µ ∈ Y sup w ∈ Γ a ,tb N | K µ ( w ) || w | . Note that ǫ ( t ) → t → ∞ using Lemma 2.6. Let z ∈ Γ a ,tb ,tc and w ∈ Γ a ,tb ,tc . Note thatIm( z − N K µ ( w )) | z − N K µ ( w ) | ≥ Im( z ) − ǫ ( t ) | w || z | + ǫ ( t ) | w | ≥ Im( z ) − ǫ ( t )( tc /tb ) Im( z ) | z | + ǫ ( t )( tc /tb ) | z | = b − ǫ ( t ) c b + ǫ ( t ) c Im z | z | ≥ b − ǫ ( t ) c b + ǫ ( t ) c a , REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 9 where we have used the fact that | z | ≥ Im z ≥ tb . Since ǫ ( t ) →
0, we have for sufficiently large t that b − ǫ ( t ) c b + ǫ ( t ) c a ≥ a . Next, note that Im( z − N K µ ( w )) ≥ Im( z ) − ǫ ( t ) | w | ≥ t [ b − ǫ ( t ) c ] . This will be ≥ tb provided that t is large enough that b − ǫ ( t ) c ≥ b . Finally, | z − N K µ ( w ) | ≤ | z | + ǫ ( t ) | w | ≤ tc + ǫ ( t ) tc = t ( c + ǫ ( t ) c ) . This will be ≤ tc provided that t is large enough that c + ǫ ( t ) c ≤ c . (cid:3) Proof of Theorem 4.1.
First, let us prove the uniqueness claim. Note that if T is a finite tree of depth d , then (4.1) expresses K ⊞ T ( µ ,...,µ N ) in terms of the branches of T , which are trees of depth at most d −
1. Therefore, by induction, ⊞ T ( µ , . . . , µ N ) is uniquely determined for all finite trees in Tree( N ).However, finite trees are dense in Tree( N ), so by continuity, ⊞ T ( µ , . . . , µ N ) is uniquely determinedfor every tree.To prove the existence and continuity claims, we begin more generally. Let Y be a compact subsetof P ( R ), and fix some a > a > a > , < b < b < b < c < c < c . Fix some t as in the conclusion of Lemma 4.4. We will apply the Earle-Hamilton theorem with X = C (Tree( N ) × Y N × Γ a ,tb ,tc ) , Ω = C (Tree( N ) × Y N × Γ a ,tb ,tc , (Γ a ,tb ,tc ) ◦ ) . To check that Ω is open, note that because Tree( N ) × Y N × Γ a ,tb ,tc , any continuous function f fromthis space into (Γ a ,tb ,tc ) ◦ will have compact image, hence the image will be separated by a positivedistance δ from C \ Γ a ,tb ,tc , and then C (Tree( N ) × Y N × Γ a ,tb ,tc , (Γ a ,tb ,tc ) ◦ ) contains the ballof radius δ/ f in C (Tree( N ) × Y N × Γ a ,tb ,tc ).Now let F : Ω → X be given by F ( f )( T , µ , . . . , µ N , z ) = z − X j ∈T ∩ [ N ] K µ j ( f (br j ( T ) , µ , . . . , µ N , z )) . Because br j is continuous, it is straightforward to check that F ( f ) is continuous, hence is an elementof X . Because K µ j is holomorphic, it follows that F is a holomorphic function Ω → X . Indeed, itsuffices to check that for each j the holomorphicity of the map F j given by F j ( f )( T , µ , . . . , µ N , z ) = j ∈T K µ j ( f (br j ( T ) , µ , . . . , µ N , z )) . Letting Tree(
N, j ) = {T ∈
Tree( N ) : j ∈ T } , we can write F j as the composition of the followingmaps: • The map C (Tree( N ) × Y N × Γ a ,tb ,tc , Γ a ,tb ,tc ) → C (Tree( N, j ) × Y N × Γ a ,tb ,tc , Γ a ,tb ,tc )given by precomposition in the T -coordinate with br j : Tree( N, j ) → Tree( N ). This is therestriction of a linear transformation C (Tree( N ) × Y N × Γ a ,tb ,tc ) → C (Tree( N, j ) × Y N × Γ a ,tb ,tc ), hence is holomorphic. • Pointwise application of K µ j , which maps C (Tree( N, j ) × Y N × Γ a ,tb ,tc , Γ a ,tb ,tc ) holo-morphically into C (Tree( N, j ) × Y N × Γ a ,tb ,tc ). • The inclusion map C (Tree( N, j ) × Y N × Γ a ,tb ,tc ) → C (Tree( N ) × Y N × Γ a ,tb ,tc ) givenby extension by zero (recall that Tree( N, j ) is clopen in Tree( N )). This map is linear, henceholomorphic.We claim that(4.2) F (Ω) ⊆ C (Tree( N ) × Y N × Γ a ,tb ,tc , Γ a ,tb ,tc ) ⊆ Ω . Fix f ∈ Ω, and fix T , µ , . . . , µ N , and z . By our choice of t (see Lemma 4.4), since µ j ∈ Y and z ∈ Γ a ,tb ,tc and f (br j ( T ) , µ , . . . , µ N , z ) ∈ Γ a ,tb ,tc , we have z − N K µ j ( f (br j ( T ) , µ , . . . , µ N , z )) ∈ Γ a ,tb ,tc . Now because Γ a ,tb ,tc is convex and contains z , the point z − X j ∈T ∩ [ N ] K µ j ( f (br j ( T ) , µ , . . . , µ N , z )) = | [ N ] ∩ T | N z + X j ∈ [ N ] ∩T N (cid:0) z − N K µ j ( f (br j ( T ) , µ , . . . , µ N , z )) (cid:1) is in Γ a ,tb ,tc . Therefore, F ( f )( T , µ , . . . , µ N , z ) ∈ Γ a ,tb ,tc , demonstrating (4.2).Now Γ a ,tb ,tc is separated by a positive distance δ from Γ ca ,tb ,tc . This implies that C (Tree( N ) × Y N × Γ a ,tb ,tc , Γ a ,tb ,tc ) is separated by δ from the complement of Ω = C (Tree( N ) × Y N × Γ a ,tb ,tc , (Γ a ,tb ,tc ) ◦ ). Therefore, the Earle-Hamilton theorem applies and there is a unique f ∈ Ωthat satisfies F ( f ) = f . Moreover, the iterates F ◦ n ( z ) (where z represents the constant function withvalue z ) converge to f in C (Tree( N ) × Y N × Γ a ,tb ,tc , (Γ a ,tb ,tc ) ◦ ) as n → ∞ .Now we can prove the existence claim. Fix µ , . . . , µ N . In the foregoing argument, we can take Y = { µ , . . . , µ N } , which is clearly compact. Let F (0) T ,µ ,...,µ N ( z ) = z and F ( n +1) T ,µ ,...,µ N ( z ) = z − X j ∈ [ N ] ∩T K µ j ( F ( n )br j ( T ) ,µ ,...,µ N ( z )) . By a straightforward induction argument, F ( n +1) T ,µ ,...,µ N ( z ) is well-defined and is a holomorphic mapfrom the upper half-plane to itself. Moreover, F ( n ) T ,µ ,...,µ N | Γ a ,tb ,tc = F ◦ n ( z )( T , µ , . . . , µ N , · ) . Hence, the preceding argument shows that F ( n ) T ,µ ,...,µ N converges uniformly on Γ a ,tb ,tc as n → ∞ .Because Hol( H , H ) is a normal family when the target space is viewed as a subset of the Riemannsphere, it follows that F ( n ) T ,µ ,...,µ N converges locally uniformly on all of H as n → ∞ to some function F T ,µ ,...,µ N , which must map into H by Hurwitz’s theorem. The identity F T ,µ ,...,µ N ( z ) = z − X j ∈ [ N ] ∩T K µ j ( F br j ( T ) ,µ ,...,µ N ( z ))holds on Γ a ,tb ,tc by the foregoing argument, and hence it holds on all of H by the identity theorem.Next, we argue that F T ,µ ,...,µ N is the F -transform of some probability measure µ . By Nevanlinna’stheorem, it suffices to show that F T ,µ ,...,µ N ( it ) /t → i as t → + ∞ on the positive real axis. For thispurpose, let us forget the original values of a j , b j , c j and t . Given a neighborhood U of i , we maychoose a > a > a > < b < b < b < c < c < c such thatΓ a ,b ,c ⊆ U. If t is sufficiently large, then the foregoing argument shows that F T ,µ ,...,µ N (Γ a ,tb ,tc ) ⊆ Γ a ,tb ,tc since the fixed point of F must clearly be in F (Ω). In particular, since it ∈ t Γ a ,b ,c = Γ a ,tb ,tc , weget F T ,µ ,...,µ N ( it ) ∈ Γ a ,tb ,tc = t Γ a ,b ,c and hence F T ,µ ,...,µ N ( it ) /t ∈ U . Thus, F T ,µ ,...,µ N is the F -transform of some probability measure ⊞ T ( µ , . . . , µ N ). This concludes the existence claim.Finally, we must show joint continuity of ( T , µ , . . . , µ N ) ⊞ T ( µ , . . . , µ N ). Since P ( R ) is metriz-able by Prokhorov’s theorem, it suffices to show sequential continuity, which in turn will follow ifwe show that the map is continuous on Tree( N ) × Y N for every compact Y ⊆ P ( R ). Fix constants a > a > a > < b < b < b < c < c < c and let t be as in Lemma 4.4. Then by theprevious argument involving the Earle-Hamilton theorem, the mapTree( N ) × Y N × Γ a ,tb ,tc → Γ a ,tb ,tc : ( T , µ , . . . , µ N , z ) F T ,µ ,...,µ N ( z )is jointly continuous, due to the definition of the set Ω. Since the domain of this function is compact,it is uniformly continuous, and henceTree( N ) × Y N → C (Γ a ,tb ,tc , Γ a ,tb ,tc ) : ( T , µ , . . . , µ N ) F T ,µ ,...,µ N | Γ a ,tb ,tc REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 11 is continuous. Because Hol( H , H ) is a normal family, uniform convergence on Γ a ,tb ,tc of a sequence F n in Hol( H , H ) to some F ∈ Hol( H , H ) implies local uniform convergence F n → F on all of H . Hence,we have continuity of the mapTree( N ) × Y N → Hol( H , H ) : ( T , µ , . . . , µ N ) F T ,µ ,...,µ N . But by Lemma 2.2, this is equivalent to continuity of ( T , µ , . . . , µ N ) ⊞ T ( µ , . . . , µ N ), which iswhat we wanted to prove. (cid:3) Corollary 4.5.
The convolution operation defined in Theorem 4.1 in the case of compactly supportedmeasures agrees with the one defined in [27] for B = C .Proof. Let
T ∈
Tree( N ), and let µ , . . . , µ N be probability measures supported in [ − R, R ]. Thepaper [27] took the viewpoint of treating the measures as positive linear functionals on the polynomialalgebra, which is equivalent in the case of compactly supported measures. The convolution operationin [27] was shown to satisfy (4.1); see section 6.3, equation (6.3) in [27]. Now the T -free convolutionof µ , . . . , µ N is supported in [ − N R, N R ], and the moments depend continuously on T ; see [27, § µ , . . . , µ N satisfies the fixed point equation and continuityproperty of Theorem 4.1, so it agrees with the convolution defined in that theorem. (cid:3) Here are a few simple cases of convolution operations that we will use later.
Example 4.6.
Let T = {∅} ∈ Tree( N ). Then K ⊞ T ( µ ,...,µ N ) ( z ) = 0because it the sum over an empty index set. Hence, ⊞ T ( µ , . . . , µ N ) = δ . Example 4.7.
Let T N, bool = {∅} ∪ [ N ] ∈ Tree( N ). Then br j ( T N, bool ) = {∅} . Therefore, K ⊞ T N, bool ( µ ,...,µ N ) ( z ) = N X j =1 K µ j ( z − K δ ( z )) = N X j =1 K µ j ( z ) . The convolution ⊞ T N, bool ( µ , . . . , µ N ) is called the boolean convolution of µ , . . . , µ N and it is com-monly denoted U Nj =1 µ j or µ ⊎ · · · ⊎ µ N (see [43]). Boolean convolution corresponds to addition of the K -transforms. Hence, the binary boolean convolution operation ⊎ is commutative and associative.Since the boolean convolution is independent of the order of the measures, we may unambiguouslywrite U s ∈ S µ s where S is a finite set. Example 4.8.
Let T orth = {∅ , , } . Then ⊞ T orth ( µ , µ ) is called the orthogonal convolution and isdenoted µ ⊢ µ (see [32]). Note that br ( T orth ) = {∅ , } and br ( T orth ) = {∅} and br (br ( T orth )) = {∅} . Therefore, K µ ⊢ µ ( z ) = K µ ( z − K µ ( z )) = K µ ◦ F µ ( z ) . Remark . The fixed point equation (4.1) can be expressed alternatively in terms of the booleanand orthogonal convolution as(4.3) ⊞ T ( µ , . . . , µ N ) = ] j ∈ [ N ] ∩T (cid:0) µ j ⊢ ⊞ br j ( T ) ( µ , . . . , µ N ) (cid:1) . Iterating this formula enables us to express the convolution associated to any finite tree in terms ofthe boolean and orthogonal convolutions. The case of compactly supported measures was alreadydone in [27, § Example 4.10.
Let T , mono = {∅ , , , } . Then ⊞ T , mono ( µ , µ ) is called the monotone convolution of µ and µ and is denoted µ ⊲ µ (see [36, 37]). Computing iteratively with (4.1) yields(4.4) µ ⊲ µ = ( µ ⊢ µ ) ⊎ µ
22 ETHAN DAVIS, DAVID JEKEL, AND ZHICHAO WANG or equivalently K µ ⊲ µ ( z ) = K µ ( z − K µ ( z )) + K µ ( z ) , which implies that F µ ⊲ µ = F µ ◦ F µ . In the next section, we will use two more simple identitiesrelating boolean, monotone, and orthogonal convolution. First,(4.5) λ ⊢ ( µ ⊲ ν ) = ( λ ⊢ µ ) ⊢ ν, holds because K λ ◦ ( F µ ◦ F ν ) = ( K λ ◦ F µ ) ◦ F ν . Second,(4.6) N ] j =1 µ j ⊢ ν = N ] j =1 ( µ j ⊢ ν )holds because ( P Nj =1 K µ j ) ◦ F ν = P Nj =1 K µ j ◦ F ν . Remark . Let us explain the connection in the free case to earlier work such as [13, 32, 9] moreprecisely. Let T , free consist of all alternating strings on { , } , let T sub = {∅ , , , , . . . } , and let T † sub = {∅ , , , , . . . } . By computing the branches of T , free , T sub and T † sub and applying (4.3), weobtain ⊞ T , free = ( µ ⊢ ⊞ T † sub ( µ, ν )) ⊎ ( ν ⊢ ⊞ T sub ( µ, ν )) ⊞ T sub ( µ, ν ) = µ ⊢ ⊞ T † sub ( µ, ν ) ⊞ T † sub ( µ, ν ) = µ ⊢ ⊞ T sub ( µ, ν ) . Therefore, when computing the free convolution by the fixed-point iteration, one only has to workwith the three trees T , free , T sub , and T † sub . One also deduces from these equations that ⊞ T , free ( µ, ν ) = ⊞ T sub ( µ, ν ) ⊎ ⊞ T † sub ( µ, ν ). Thus, to compute ⊞ T , free ( µ, ν ), it suffices to compute ⊞ T sub ( µ, ν ) and ⊞ T † sub ( µ, ν ) by a pair of fixed-point equations as in [9].The convolution operation associated to T , free is called the free convolution µ ⊞ ν . The convolutionoperation associated to T , sub is called the subordination convolution and is denoted µ i ν . Further-more, it is easy to check (and follows from Proposition 5.1 below) that ⊞ T † sub ( µ, ν ) = ⊞ T sub ( ν, µ ) = ν i µ . Thus, µ ⊞ ν = ( µ i ν ) ⊎ ( ν i µ ) µ i ν = µ ⊢ ( ν i µ ) ν i µ = ν ⊢ ( µ i ν ) . This implies µ ⊞ ν = ( µ i ν ) ⊎ ( ν ⊢ ( µ i ν )) = ν ⊲ ( µ i ν ) . Hence, F µ ⊞ ν = F ν ◦ F µ i ν , and similarly, F µ ⊞ ν = F µ ◦ F ν i µ . Together with F µ ⊞ ν ( z ) = F µ i ν ( z ) + F ν i µ ( z ) − z , this implies that F − µ ( z ) + F − ν ( z ) − z = F − µ ⊞ ν ( z ) whenever these functions are defined.In other words, the definition of µ ⊞ ν here agrees with that of [13].5. Convolution and the operad structure
In this section, we describe how the convolution operation of Theorem 4.1 relates to the operationsin the operad Tree. We start out with two propositions that prove permutation-equivariance as wellas more general convolution identities. We remark that Propositions 5.1 and 5.2 imply that all thesame convolution identities as in [27, §
6] hold for arbitrary probability measures on R since the onlyingredients needed in the proofs are the relations (5.1) and (5.2). Proposition 5.1.
Let ψ : [ N ] → [ N ′ ] be surjective. Let Tree( ψ ) be the set of trees T ∈
Tree( N ) suchthat ψ ∗ ( s ) is alternating for every s ∈ T and such that ψ ∗ | T is injective. Let µ , . . . , µ N ′ ∈ P ( R ) and T ∈
Tree( ψ ) . Then (5.1) ⊞ T ( µ ψ (1) , . . . , µ ψ ( N ) ) = ⊞ ψ ∗ ( T ) ( µ , . . . , µ N ′ ) . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 13
In the case of compactly supported measures, this proposition follows from [27, Corollary 5.15].One can deduce the general case by continuity because compactly supported measures are dense in P ( R ). But below we give an alternative self-contained argument directly from Theorem 4.1. Proof of Proposition 5.1.
Note that Tree( ψ ) is closed under taking branches and rooted subtrees.Therefore, finite trees are dense in Tree( ψ ), so by continuity, it suffices to prove (5.1) when T is finite.We proceed by induction on the depth of T . When the depth of T is zero, (5.1) holds because bothsides are δ . For the inductive step, consider a finite tree T of depth d . By Theorem 4.1, ⊞ T ( µ ψ (1) , . . . , µ ψ ( N ) ) = ] j ∈ [ N ] ∩T (cid:0) µ ψ ( j ) ⊢ ⊞ br j ( T ) ( µ ψ (1) , . . . , µ ψ ( N ) ) (cid:1) . Since ψ ∗ | T is injective, each neighbor i of the ∅ in ψ ∗ ( T ) is the image of a single neighbor j of ∅ in T . Moreover, br i ( ψ ∗ ( T )) = ψ ∗ (br j ( T )). Since br j ( T ) has depth strictly less than d , the inductivehypothesis implies that ⊞ br j ( T ) ( µ ψ (1) , . . . , µ ψ ( N ) ) = ⊞ ψ ∗ (br j ( T )) ( µ , . . . , µ N ′ ) . Therefore, the above expression equals ⊞ T ( µ ψ (1) , . . . , µ ψ ( N ) ) = ] i ∈ [ N ′ ] ∩ ψ ∗ ( T ) (cid:0) µ i ⊢ ⊞ br i ( ψ ∗ ( T )) ( µ , . . . , µ N ′ ) (cid:1) = ⊞ ψ ∗ ( T ) ( µ , . . . , µ N ′ ) , which completes the inductive step and hence the proof. (cid:3) Since any function is the composition of a surjection and injection, to understand the general caseof ψ : [ N ] → [ N ′ ], all that is left is to handle the injective case. Due to permutation-equivariance,we may restrict our attention to the canonical inclusion [ N ] → [ N ′ ] for N ′ > N sending j to itself inorder to simplify notation. Proposition 5.2.
Let
N < N ′ . Let ι : [ N ] → [ N ′ ] be the canonical inclusion. Then for T ∈
Tree( N ) and µ , . . . , µ N ′ ∈ P ( R ) , we have (5.2) ⊞ T ( µ , . . . , µ N ) = ⊞ ι ∗ ( T ) ( µ , . . . , µ N ′ ) . Proof.
Let M ( T ) = ⊞ ι ∗ ( T ) ( µ , . . . , µ N ′ ). Note that M ( T ) = ] j ∈ [ N ′ ] ∩ ι ∗ ( T ) µ j ⊢ ⊞ br j ( ι ∗ ( T )) ( µ , . . . , µ N ′ ) . But [ N ′ ] ∩ ι ∗ ( T ) = ι ([ N ] ∩ T ) and br j ( ι ∗ ( T )) = ι ∗ (br j ( T )). Thus, M ( T ) = ] j ∈ [ N ] ∩T µ j ⊢ ⊞ ι ∗ (br j ( T )) ( µ , . . . , µ N ′ ) = ] j ∈ [ N ] ∩T µ j ⊢ M (br j ( T )) . Thus, M ( T ) satisfies the fixed-point equation (4.1). It also depends continuously on T since T 7→ ι ∗ ( T ) is isometric. Therefore, by Theorem 4.1, M ( T ) = ⊞ T ( µ , . . . , µ N ). (cid:3) The next theorem shows that the convolution operation ⊞ T respects operad composition. Theorem 5.3.
Let
T ∈
Tree( k ) and T ∈ Tree( n ) , . . . , T k ∈ Tree( n k ) . Let N = n + · · · + n k . Foreach j ∈ [ k ] and i ∈ [ n j ] , let µ i,j ∈ P ( R ) . Then have ⊞ T ( T ,..., T k ) ( µ , , . . . , µ ,n , . . . . . . , µ k, , . . . , µ k,n k ) = ⊞ T ( ⊞ T ( µ , , . . . , µ ,k ) , . . . , ⊞ T k ( µ k, , . . . , µ k,n k )) . In the case of compactly supported measures, this result follows immediately from [27, Corollary5.13] taking B = C . Because compactly supported measures are dense in P ( R ) and because ofcontinuity of the convolution operations in Theorem 4.1, it follows that the identity holds for allmeasures in P ( R ). Although this argument is satisfactory, we will also present an alternative proofdirectly from Theorem 4.1 that is self-contained and elucidates the connection between the fixed-pointequation in Theorem 4.1, the operad structure, and the branch maps. Proof of Theorem 5.3.
First, we prove the case of the theorem where T = T , mono . In other words,we want to establish the identity(5.3) ⊞ T , mono ( T , T ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) = ⊞ T ( µ , , . . . , µ ,n ) ⊲ ⊞ T ( µ , , . . . , µ ,n )for T ∈ Tree( m ) and T ∈ Tree( n ) and for probability measures µ j,i . Note that both sides dependcontinuously on T and T , using continuity of the operad composition in Tree and continuity of theconvolution operation in Theorem 4.1. Therefore, it suffices to prove the statement when T and T are finite trees.We proceed by induction on the depth of T plus the depth of T . In the base case of combineddepth 0, we have T = {∅} , and hence T , mono ( T , T ) = T and ⊞ T ( µ , . . . , µ m ) = δ , so the claimholds.For the inductive step, consider trees T and T with combined depth d . Let T ′ = T , mono ( T , T ).Let µ = ⊞ T ( µ , , . . . , µ ,n ) and µ = ⊞ T ( µ , , . . . , µ ,n ). Note that[ n + n ] ∩ T ′ = ι ([ n ] ∩ T ) ⊔ ι ([ n ] ∩ T ) . Thus, by equation (4.3), ⊞ T ′ ( µ , , . . . , µ ,n , µ , , . . . , µ ,ν ) = ] i ∈ [ n ] ∩T (cid:16) µ ,i ⊢ ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) (cid:17) ⊎ ] i ∈ [ n ] ∩T (cid:16) µ ,i ⊢ ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) (cid:17) . Now by Lemma 3.9, br ι ( i ) ( T ′ ) = br ι ( i ) ( T , mono ( T , T ))= ( ψ ) ∗ [ T , mono (br i ( T ) , br ( T , mono )( T , T ))]= ( ψ ) ∗ [ T , mono (br i ( T ) , ( ι ) ∗ ( T ))]= T , mono (br i ( T ) , T ) , where ψ : [2 n + n ] → [ n + n ] is the map sending the first n points monotonically onto [ n ] andthe last n + n points monotonically onto [ n + n ]. Similarly,br ι ( i ) ( T ′ ) = br ι ( i ) ( T , mono ( T , T ) = ( ψ ) ∗ [ T , mono (br i ( T ) , br ( T , mono )( T , T ))] = ( ι ) ∗ (br i ( T )) . where ψ : [ n + 2 n ] → [ n + n ] sends the first n coordinates monotonically onto ι ([ n ]) and thelast n + n coordinates monotonically onto [ n + n ]. Therefore, using the induction hypothesis, ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , ν , , . . . , µ ,n ) = ⊞ br i ( T ) ( µ , , . . . , µ ,n ) ⊲ ⊞ T ( µ , , . . . , µ ,n )= ⊞ br i ( T ) ( µ , , . . . , µ ,n ) ⊲ µ , and by applying Proposition 5.2 to ι , ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) = ⊞ ( ι ) ∗ [br i ( T )] ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) . = ⊞ br i ( T ) ( µ , , . . . , µ ,n ) . Therefore, using (4.5) and (4.6), ] i ∈ [ n ] ∩T (cid:16) µ ,i ⊢ ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) (cid:17) = ] i ∈ [ n ] ∩T (cid:0) µ ,i ⊢ (cid:0) ⊞ br i ( T ) ( µ , , . . . , µ ,n ) ⊲ µ (cid:1)(cid:1) = ] i ∈ [ n ] ∩T (cid:0) µ ,i ⊢ (cid:0) ⊞ br i ( T ) ( µ , , . . . , µ ,n ) ⊲ µ (cid:1)(cid:1) = ] i ∈ [ n ] ∩T (cid:0)(cid:0) µ ,i ⊢ ⊞ br i ( T ) ( µ , , . . . , µ ,n ) (cid:1) ⊢ µ (cid:1) REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 15 = ] i ∈ [ n ] ∩T (cid:0) µ ,i ⊢ ⊞ br i ( T ) ( µ , , . . . , µ ,n ) (cid:1) ⊢ µ = µ ⊢ µ . Similarly, ] i ∈ [ n ] ∩T (cid:16) µ ,i ⊢ ⊞ br ι i ) ( T ′ ) ( µ , , . . . , µ ,n , µ , , . . . , µ ,n ) (cid:17) = ] i ∈ [ n ] ∩T (cid:0) µ ,i ⊢ ⊞ br i ( T ) ( µ , , . . . , µ ,n ) (cid:1) = µ . Therefore, ⊞ T ′ ( µ , , . . . , µ ,n , µ , , . . . , µ ,ν ) = ( µ ⊢ µ ) ⊎ µ = µ ⊲ µ as desired, which completes the inductive step.Finally, we begin the main argument to prove the general case of the theorem. Let T , T , . . . , T k and µ j,i be as in the theorem statement. Let µ j = ⊞ T j ( µ j, , . . . , µ j,n j ) . Let M ( T ) = ⊞ T ( T ,..., T k ) ( µ , , . . . , µ ,n , . . . . . . , µ k, , . . . , µ k,n k ) . Note that
T 7→ M ( T ) is continuous because composition and convolution are continuous. Thus, byTheorem 4.1, to show that M ( T ) = ⊞ T ( µ , . . . , µ k ), it suffices to show that(5.4) M ( T ) = ] j ∈ [ k ] ∩T ( µ j ⊢ M (br j ( T ))) . Let T ′ = T ( T , . . . , T k ). Let ψ j : [ n j + N ] → [ N ] map the first n j elements monotonically onto ι j ([ n j ])and the last N elements monotonically onto [ N ]. Applying (4.3), Lemma 3.9, Proposition 5.1, (5.3),and (4.5), M ( T ) = ] j ∈ [ k ] ,i ∈ [ n j ] ,ι j ( i ) ∈ [ N ] ∩T ′ (cid:16) µ j,i ⊢ ⊞ br ιj ( i ) ( T ( T ,..., T k )) ( µ , , . . . , µ k,n k ) (cid:17) = ] j ∈ [ k ] ,i ∈ [ n j ] ,ι j ( i ) ∈ [ N ] ∩T ′ (cid:0) µ j,i ⊢ ⊞ ( ψ j ) ∗ [ T , mono (br i ( T j ) , br j ( T )( T ,..., T k ))] ( µ , , . . . , µ k,n k ) (cid:1) = ] j ∈ [ k ] ∩T ] i ∈ [ n j ] ∩T j (cid:0) µ j,i ⊢ ⊞ T , mono (br i ( T j ) , br j ( T )( T ,..., T k )) ( µ j, , . . . , µ j,n j , µ , , . . . , µ k,n k ) (cid:1) = ] j ∈ [ k ] ∩T ] i ∈ [ n j ] ∩T j (cid:0) µ j,i ⊢ ( ⊞ br i ( T j ) ( µ j, , . . . , µ j,n j ) ⊲ ⊞ br j ( T )( T ,..., T k ) ( µ , , . . . , µ k,n k )) (cid:1) = ] j ∈ [ k ] ∩T ] i ∈ [ n j ] ∩T j (cid:0) ( µ j,i ⊢ ⊞ br i ( T j ) ( µ j, , . . . , µ j,n j )) ⊢ M (br j ( T )) (cid:1) = ] j ∈ [ k ] ∩T ( µ j ⊢ M (br j ( T ))) , which demonstrates (5.4) and hence finishes the proof. (cid:3) Our final observation is that the T -free convolution of several copies of the same measure dependsonly on the isomorphism class of T . Lemma 5.4.
Suppose T ∈ Tree( N ) and T ∈ Tree( N ) . If T ∼ = T , then ⊞ T ( µ, . . . , µ ) = ⊞ T ( µ, . . . , µ ) for all µ ∈ P ( R ) . Proof.
Clearly, if T and T are isomorphic, then their truncations T ( k )1 and T ( k )2 to depth k are also iso-morphic for every k . Since T ( k )1 → T and T ( k )2 → T in Tree( N ) and Tree( N ) respectively, and sincethe convolution operations are continuous, it suffices to show that ⊞ T ( k )1 ( µ, . . . , µ ) = ⊞ T ( k )2 ( µ, . . . , µ ).Therefore, to prove the lemma, it suffices to prove the case where T and T are finite trees. Weproceed by induction on the depth, the depth-0 case being trivial. Let φ : T → T be an isomorphism.By Observation 3.11, φ defines a bijection [ N ] ∩ T → [ N ] ∩ T , and br j ( T ) ∼ = br φ ( j ) ( T ) for each j ∈ [ N ] ∩ T . Since the branches have strictly smaller depth, we may apply the induction hypothesisto them. Hence, ⊞ T ( µ, . . . , µ ) = ] j ∈ [ N ] ∩T µ ⊢ ⊞ br j ( T ) ( µ, . . . , µ )= ] j ′ ∈ [ N ] ∩T µ ⊢ ⊞ br j ′ ( T ) ( µ, . . . , µ ) = ⊞ T ( µ, . . . , µ ) , which completes the inductive step and hence the proof. (cid:3) A general limit theorem
Bercovici and Pata [12, Theorem 6.3] showed a bijection between limit theorems for classical, free,and boolean convolution in the following sense: Given a sequence of measures µ ℓ on R and k ℓ ∈ N tending to infinity, µ ∗ k ℓ ℓ converges weakly as ℓ → ∞ if and only if µ ⊞ k ℓ ℓ converges if and only if µ ⊎ k ℓ ℓ converges. Theorem 6.1 will generalize one direction of this result to trees T with n ( T ) > T .Applications of this result as well as open questions will be discussed in § T ∈
Tree( N ) and µ ∈ P ( R ), let ⊞ T ( µ ) := ⊞ T ( µ, . . . , µ | {z } N times ) . We also use boolean convolution powers defined as follows: For c > µ ∈ P ( R ), let µ ⊎ c be givenby K µ ⊎ c = cK µ . For each µ , such a measure µ ⊎ c exists because a function K is the K -transform of a measure if andonly if K maps H to − H and K ( z ) /z → z → ∞ in H non-tangentially. Clearly, µ ⊎ c is well-definedsince a measure is uniquely determined by its K -transform. If N ∈ N , then U Nj =1 µ = µ ⊎ N . We alsohave ( µ ⊎ c ) ⊎ c = µ ⊎ c c . Recall also Definition 3.12 and Lemma 3.16. Theorem 6.1.
Let ( µ ℓ ) ℓ ∈ N be a sequence in P ( R ) and let ( k ℓ ) ℓ ∈ N be a sequence of natural numberstending to ∞ . Let N ∈ N and T ∈
Tree( N ) with n ( T ) > . If µ ⊎ n ( T ) kℓ ℓ converges to some probabilitymeasure ν as ℓ → ∞ , then ⊞ T ◦ kℓ ( µ ℓ ) converges as ℓ → ∞ to some probability measure BP ( T , ν ) onlydepending on T and ν . Moreover, the convergence is uniform overall T ∈
Tree( N ) with n ( T ) > . Our proof relies on the following result, which gives certain continuity estimates for the T -freeconvolution operations that are independent of N . Theorem 6.2.
For N ∈ N , we define Φ N : Tree( N ) × [0 , × P ( R ) → P ( R ) : by Φ N ( T , c, µ ) := ( ⊞ T ( µ ⊎ c/N , . . . , µ ⊎ c/N ) ⊎ /c , c ∈ (0 , ,µ ⊎ n ( T ) N , c = 0 . The map Φ N satisfies the fixed-point equation (6.1) K Φ N ( T ,c,µ ) ( z ) = 1 N X j ∈ [ N ] ∩T K µ ( z − cK Φ N (br j ( T ) ,c,µ ) ( z )) . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 17
Moreover, the maps (Φ N ) N ∈ N have the following equicontinuity property: For each compact Y ⊆ P ( R ) and ǫ > , there exists δ > such that for all N , for all T , T ∈ Tree( N ) and c , c ∈ [0 , and µ ∈ Y and ν ∈ P ( R ) , if ρ N ( T , T ) + | c − c | + d L ( µ, ν ) < δ , then d L (Φ N ( T , c , µ ) , Φ N ( T , c , ν )) < ǫ .Proof. To check (6.1) for c >
0, observe that K Φ N ( T ,c,µ ) ( z ) = 1 c K ⊞ T ( µ ⊎ c/N ,...,µ ⊎ c/N ) ( z )= X j ∈ [ N ] ∩T c K µ ⊎ c/N ( z − K ⊞ br j ( T ) ( µ ⊎ c/N ,...,µ ⊎ c/N ) ( z ))= 1 N X j ∈ [ N ] ∩T K µ ( z − cK Φ N (br j ( T ) ,c,µ ) ( z )) . The case c = 0 is immediate and left to the reader.Now we turn to the claim about continuity. We will show below that the family (Φ N ) N ∈ N isuniformly equicontinuous on Tree( N ) × [0 , × Y . By this, we mean more precisely that the functionsare uniformly continuous with a modulus of continuity that is independent of N ; even though thedomains are different, we have fixed a metric ρ N for each Tree( N ) from the beginning. This claimabout equicontinuity for each compact Y is enough to finish the proof. Indeed, if the conclusion ofthe theorem failed, then there would be a compact set Y and ǫ > k >
0, thereexist µ k ∈ Y and ν k ∈ P ( R ) and N k ∈ N and T k , T ′ k ∈ Tree( N k ) and c k , c ′ k ∈ [0 ,
1] such that ρ N ( T k , T ′ k ) + | c k − c ′ k | + d L ( µ k , ν k ) < /k, d L (Φ N k ( T k , c k , µ k ) , Φ N k ( T ′ k , c ′ k , ν k )) ≥ ǫ. Then Y ′ = Y ∪ { ν k : k ∈ N } would be compact, and the above conditions would contradict theequicontinuity on Tree( N ) × [0 , × Y ′ .As before, the strategy is to reframe (6.1) as a fixed-point equation for some analytic function F on a Banach space X and apply the Earle-Hamilton theorem. Fix Y ⊆ P ( R ) compact, and fix a > a > a > , < b < b < b < c < c < c , and let t be as in the conclusion of Lemma 2.2. Let X to be the space of sequences ( f N ) N ∈ N where f N : Tree( N ) × [0 , × Y × Γ a ,tb ,tc → C and where the f N ’s are uniformly bounded and uniformlyequicontinuous. Let Ω = ( ( f N ) N ∈ N ∈ X : [ N ∈ N Ran( f N ) ⊆ (Γ a ,tb ,tc ) ◦ ) , where Ran( f N ) denotes the range (image) of f N . Note that Ω is open in X . Define F : Ω → X by F (( f N ) N ∈ N ) := ( g N ) N ∈ N , where g N ( T , c, µ, z ) = z − N X j ∈ [ N ] ∩T K µ ( f N (br j ( T ) , c, µ, z )) . We must check that ( g N ) N ∈ N is actually in X , that F is analytic, and F (Ω) is separated by a positivedistance from Ω c . First, to show that ( g N ) N ∈ N is equicontinuous, one combines the following facts:(1) the modulus of continuity of the map br j (on its domain) is independent of N since it is e − Lipschitz;(2) the map µ K µ is uniformly continuous on Y where we use the weak topology on P ( R ) and thetopology of uniform convergence on Γ a ,tb ,tc ;(3) for µ ∈ Y , the transforms K µ are uniformly bounded on some neighborhood of Γ a ,tb ,tc andhence the maps { K µ : µ ∈ Y } are equicontinuous on Γ a ,tb ,tc .To see that ( g N ) N ∈ N is uniformly bounded, note that by our choice of t , g N ( T , c, µ, z ) will be inΓ a ,tb ,tc for every T , c , µ , and z since it is a convex combination of z and other points in Γ a ,tb ,tc as in the proof of Theorem 4.1. This same fact implies the separation of F (Ω) from Ω c . The analyticityof F is straightforward to check as in the proof of Theorem 4.1. For each N ∈ N , let f N ( T , c, µ, z ) = z . Then the Earle-Hamilton theorem implies that F ◦ n (( f N ) N ∈ N )converges as n → ∞ to the unique fixed point ( f N ) N ∈ N . As in the proof of Theorem 4.1, the iteratesthemselves are F -transforms of measures, and therefore the convergence extends to the entire upperhalf-plane. And there is a measure Ψ N ( T , c, µ ) such that f N ( T , c, µ, z ) = F Ψ N ( T ,c,µ ) ( z ). Because( f N ) is uniformly equicontinuous, we see that (Ψ N ) N ∈ N is uniformly equicontinuous, since uniformconvergence of a sequence of F -transforms on Γ a ,tb ,tc is equivalent to weak convergence of the as-sociated sequence of measures. Finally, since Ψ N ( T , c, µ ) ⊎ c depends continuously on T and satisfiesthe fixed point equation defining ⊞ T ( µ ⊎ ( c/N ) , . . . , µ ⊎ ( c/N ) ) = Φ N ( T , c, µ ), Theorem 4.1 implies thatΨ N = Φ N . Hence, the equicontinuity properties proved for Ψ N hold for Φ N . (cid:3) Theorem 6.3.
Let
T ∈
Tree( N ) with n ( T ) > . For µ ∈ P ( R ) , we have existence of the limit BP ( T , µ ) := lim k →∞ ⊞ T ◦ k ( µ ⊎ n ( T ) k ) . Moreover, the convergence is uniform on {T ∈
Tree( N ) : n ( T ) > } × Y for every compact subset of P ( R ) , and hence the BP is a continuous map {T ∈ Tree( N ) : n ( T ) > } × P ( R ) → P ( R ) .Remark . We call the map “ BP ”in honor of Bercovici and Pata. Proof.
Let Tree(
N, n ) = {T ∈
Tree( N ) : n ( T ) = n } , which is a clopen subset of Tree( N ). Note that {T ∈ Tree( N ) : n ( T ) > } = S Nn =2 Tree(
N, n ).Fix n ∈ { , . . . , N } , and let Y be a compact subset of P ( R ), and we will show uniform convergenceof ⊞ T ◦ k ( µ ⊎ nk ) on Tree( N, n ) × Y . This of course is enough to imply continuity of the limit function.And to show uniform continuity, it suffices to show that the sequence is uniformly Cauchy respect tothe L´evy distance d L . Fix an integer M ≥ N − n − ≥ . For each
T ∈
Tree(
N, n ), we have by Lemma 3.16 and Observation 3.13 that m ( T ◦ k ) + 1 = m ( T ) n k − n − ≤ N − n − n k −
1) + 1 ≤ N − n − n k ≤ M n k . Therefore, by Observation 3.13, there exists some tree T k ∈ Tree(
M n k ) such that T k ∼ = T ◦ k . Hence,by Lemmas 5.4 and 3.17, we have for k, ℓ ≥
1, and µ ∈ P ( R ) that ⊞ T ◦ ( k + ℓ ) ( µ ) = ⊞ T ◦ k ( ⊞ T ◦ ℓ ( µ )) = ⊞ T k ( ⊞ T ℓ ( µ )) = ⊞ T k ◦T ℓ ( µ ) . In particular, ⊞ T ◦ ( k + ℓ ) ( µ ⊎ nk + ℓ ) = ⊞ T k ( ⊞ T ℓ ( µ ⊎ nk ))= ⊞ T k (cid:18)(cid:16) ⊞ T ℓ (( µ ⊎ M ) ⊎ MnℓMnk ) ⊎ Mn k (cid:17) ⊎ Mnk (cid:19) = Φ Mn k (cid:18) T k , , Φ Mn ℓ (cid:18) T ℓ , M n k , µ ⊎ M (cid:19)(cid:19) . To show the sequence is Cauchy, fix ǫ >
0. Note that Y ⊎ M = { µ ⊎ M : µ ∈ Y } is compact since theboolean convolution power is simply a rescaling of the K -transform, and the same holds for Y ⊎ M .Thus, by Theorem 6.2, there exists η > k , for all λ ∈ Y ⊎ M and ν ∈ P ( R ), d L ( λ, ν ) < η = ⇒ d L (Φ Mn k ( T k , , λ ) , Φ Mn k ( T k , , ν )) < ǫ . Applying Theorem 6.2 again, there exists δ > ℓ ∈ N , for all λ ∈ Y ⊎ M , we have c ∈ [0 , δ ) = ⇒ d L (Φ Mn ℓ ( T ℓ , c, λ ) , Φ Mn ℓ ( T ℓ , , λ )) < η. Note that Φ Mn ℓ ( T ℓ , , µ ⊎ M ) = ( µ ⊎ M ) ⊎ nℓMnℓ = µ ⊎ M . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 19
Hence, if k > log n (1 /M η ) and µ ∈ Y and ℓ ≥
1, then d L (Φ Mn ℓ ( T ℓ , , µ ⊎ M ) , µ ⊎ M ) < η, hence d L (cid:18) Φ Mn k (cid:18) T k , , Φ Mn ℓ (cid:18) T ℓ , M n k , µ ⊎ M (cid:19)(cid:19) , Φ Mn k ( T k , , µ ⊎ M ) (cid:19) < ǫ . So for µ ∈ Y and ℓ, ℓ ′ ≥ d ℓ (cid:16) ⊞ T ◦ ( k + ℓ ) ( µ ⊎ nk + ℓ ) , ⊞ T ◦ ( k + ℓ ′ ) ( µ ⊎ nk + ℓ ′ ) (cid:17) < ǫ. Therefore, the sequence is uniformly Cauchy as desired. (cid:3)
Proof of Theorem 6.1.
Let
T ∈
Tree( N ) and µ ℓ , ν ∈ P ( R ). Suppose that ν ℓ := µ ⊎ n ( T ) kℓ ℓ → ν . Let Y ⊆ P ( R ) be a compact set containing all the measures ν ℓ . Theorem 6.3 implies uniform convergenceof ⊞ T ◦ kℓ ( λ ⊎ n ( T ) − kℓ ) → BP ( T , λ ) over λ ∈ Y as ℓ → ∞ . Since BP ( T , λ ) is continuous and because ofthe uniform convergence, we can still take limits as ℓ → ∞ with λ replaced by the sequence ν ℓ thatdepends on ℓ . Thus, ⊞ T ◦ kℓ ( µ ℓ ) = ⊞ T ◦ kℓ ( ν ⊎ n ( T ) kℓ ℓ ) → BP ( T , ν )as desired. The convergence is uniform over T ∈
Tree( N ) because the convergence in Theorem 6.3 isuniform and because of BP is uniformly continuous on Tree( N ) × Y . (cid:3) Limit theorems for classical domains of attraction
Practically speaking, Theorem 6.1 means that known limit theorems for boolean convolution carryover automatically to T -free convolution. First, we have the following central limit theorem. Below,if µ ∈ P ( R ) and c ∈ R , then c · µ denotes the dilation of µ by c , that is, the push-forward of µ by thefunction t ct . Proposition 7.1 (Central limit theorem) . Let
T ∈
Tree( N ) with n ( T ) > and let µ ∈ P ( R ) be ameasure with mean zero and variance . Let ν be the Bernoulli distribution (1 / δ − + δ ) . Then lim k →∞ n ( T ) − k/ ⊞ T ◦ k ( µ ) = BP ( T , ν ) , and the convergence is uniform in the L´evy distance over all T ∈
Tree( N ) with n ( T ) > . This will be an immediate consequence of [43, Theorem 3.4] and Theorem 6.1, once we first establishthe basic properties of dilations.
Lemma 7.2. (1) For c = 0 , have K c · µ ( z ) = cK µ ( z/c ) .(2) For T ∈
Tree( N ) , we have ⊞ T ( c · µ , . . . , c · µ N ) = c · ⊞ T ( µ , . . . , µ N ) . (3) When n ( T ) > , the map BP from Theorem 6.3 satisfies BP ( T , c · µ ) = c · BP ( T , µ ) . (4) When n ( T ) > , we have ⊞ T ( BP ( T , µ )) = BP ( T , µ ⊎ n ( T ) ) . Proof. (1) Note that G c · µ ( z ) = Z R z − ct dµ ( t ) = 1 c Z R z/c − t dµ ( t ) = 1 c G µ ( z/c ) . Hence, F c · µ ( z ) = cF µ ( z/c ) and K c · µ ( z ) = K µ ( z/c ). (2) In the case c = 0, both sides are δ . For c = 0, note that c · ⊞ T ( µ , . . . , µ N ) depends continuouslyon T and satisfies the fixed-point equation K c · ⊞ T ( µ ,...,µ N ) ( z ) = X j ∈ [ N ] ∩T K c · µ j ( z − K c · ⊞ br j ( T ) ( µ ,...,µ N ) ( z )) , hence, by Theorem 4.1, we have the desired equality.(3) From (1) it follows that ( c · µ ) ⊎ t = c · ( µ ⊎ t ) for c ∈ R and t >
0. Therefore, using (2), BP ( T , c · µ ) = lim k →∞ ⊞ T ◦ k (( c · µ ) ⊎ n ( T ) k )= lim k →∞ ⊞ T ◦ k ( c · ( µ ⊎ n ( T ) k ))= lim k →∞ c · ⊞ T ◦ k ( µ ⊎ n ( T ) k )= c · BP ( T , µ ) . (4) Observe that ⊞ T ( BP ( T , µ )) = lim k →∞ ⊞ T ( ⊞ T ◦ k ( µ ⊎ n ( T ) k ))= lim k →∞ ⊞ T ◦ ( k +1) (( µ ⊎ n ( T ) ) ⊎ n ( T ) k +1 )= BP ( T , µ ⊎ n ( T ) ) . (cid:3) Proof of Proposition 7.1.
It follows from [43, Theorem 3.4] that n ( T ) − k/ · µ ⊎ n ( T ) k = ( n ( T ) − k/ · µ ) ⊎ n ( T ) k → (1 / δ − + δ ). Therefore, the proposition follows from Theorem 6.1 and the fact that n ( T ) − k/ · ⊞ T ◦ k ( µ ) = ⊞ T ◦ k ( n ( T ) − k/ · µ ). (cid:3) Following a similar strategy as Bercovici and Pata [12], we can use Theorem 6.1 to prove analoguesof classical limit theorems associated to other stable distributions. To set the stage, we recall someterminology used in the classification of domains of attraction in classical probability theory; see [12, § Definition 7.3.
We say that two measures µ and ν are equivalent if µ = a + b · ν for some a ∈ R and b >
0. A measure µ is said to be ∗ -stable if its equivalence class is closed under the classicalconvolution operation ∗ ; ⊞ -stable is defined analogously. Definition 7.4.
A function f : [0 , ∞ ) → [0 , ∞ ) varies slowly iflim y →∞ f ( ty ) f ( y ) = 1 for t > . We say that f varies regularly with index α if f ( y ) /y α varies slowly, or equivalentlylim y →∞ f ( ty ) t α f ( y ) = 1 for t > . We make the same definitions for functions only defined on [ m, ∞ ) for some m > Definition 7.5.
We say that a measure µ belongs to C if the function y R y − y t dµ ( t ) varies slowly. Definition 7.6.
For α ∈ (0 ,
2) and θ ∈ [ − , µ belongs to C α,θ if(1) the function y R y − y t dµ ( t ) varies regularly with index 2 − α ;(2) we have lim t →∞ µ (( t, ∞ )) − µ (( −∞ , − t ) µ (( t, ∞ )) + µ (( −∞ , − t ) = θ. The following is a classical result.
REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 21
Theorem 7.7.
There exists a unique equivalence class of ∗ -stable laws in each of the sets C and C α,θ for α ∈ (0 , and θ ∈ [ − , . Let ν ∗ and ν ∗ α,θ be representatives of these equivalence classes.Then for each µ ∈ C or µ ∈ C α,θ , there exists a sequence of measures µ n ∼ µ such that µ ∗ nn → ν or µ ∗ nn → ν α,θ respectively (that is, µ is in the domain of attraction of ν ∗ or ν ∗ α,θ ). Bercovici and Pata used this theorem together with their [12, Theorem 6.3] to deduce limit lawsfor free and boolean convolution. We want to do the same thing for T -free convolution. One obstaclefor the general case is that translation of measures does not behave well with respect to T -freeconvolutions. If c + µ denotes the translation of µ by c ∈ R , then we do not have ⊞ T ( c + µ ) = n ( T ) c + ⊞ T ( µ ). For instance, the measure δ c ⊞ µ = δ c ∗ µ = µ ⊲ δ c = c + µ has K -transform equal to K µ ( z − c ) + c ; however, δ c ⊎ µ = δ c ⊲ µ has K -transform K µ ( z ) + c , and hence does not agree with c + µ .In the case α ∈ (0 , α ∈ (1 , C α,θ has finite mean, andhence we will restrict our attention to the set of measures in C α,θ with mean zero, which we denote by C α,θ . The case α = 1 is difficult because the mean may or may not be defined, and one must inevitablydeal with drift, which brings up the tricky question of translation. In Theorem 7.9, we handle thecases of C α,θ with α ∈ (0 ,
1) and C α,θ with α ∈ (1 , α = 1 and α = 2, we will deduce a less sharpresult from the classical theory and the work of Bercovici and Pata. Proposition 7.8.
For α ∈ (0 , and θ ∈ [ − , , there is a measure ν α,θ with K ν α,θ ( z ) = ( − ( i − θ tan πα )( − iz ) − α , α = 1 , θ log( − iz ) − iπ, α = 1 , for z in the upper half-plane, where we use the branch of the log function where the argument is in ( − π, π ] . For α ∈ (0 , ∪ (1 , and c > , we have c · ν α,θ = ν ⊎ c α α,θ . Moreover, for c > , we have c · ν α,θ = ( δ − θ log c ⊎ ν α,θ ) ⊎ c .Proof. Let K α,θ be the function on the right-hand side. One can verify by direct computation that K α,θ maps the upper half-plane into the lower half-plane and that K α,θ ( z ) /z → z → ∞ non-tangentially. Thus, by Corollary 2.5, K α,θ is the K -transform of some measure ν α,θ . The finalclaim follows from direct computation using Lemma 7.2 (1) and the definition of boolean convolutionpowers. (cid:3) Theorem 7.9.
Suppose that α ∈ (0 , ∪ (1 , , θ ∈ [ − , , and µ ∈ C α,θ . If α ∈ (1 , , then assume inaddition that µ has mean zero. Then there exists some φ : [0 , + ∞ ) → [0 , + ∞ ) which varies regularlywith index − /α such that for all N and for all T ∈
Tree( N ) with n ( T ) > , φ ( n ( T ) k ) · ⊞ T ◦ k ( µ ) → BP ( T , ν α,θ ) . For each ( α, θ ) and for each N , the convergence is uniform over T ∈
Tree( N ) with n ( T ) > . For examples of the distributions BP ( T , ν α,θ ), see Figures 1 and 2 at the end of the paper. Theproof of the theorem relies on the following characterization of C α,θ in terms of the Cauchy transform,which is due to Bercovici and Pata. Proposition 7.10 ([12, Proposition 5.10-5.11]) . Let α ∈ (0 , ∪ (1 , , θ ∈ [ − , , and µ ∈ P ( R ) . Inthe case α > , assume in addition that µ has mean zero. Then µ ∈ C α,θ if and only if there existssome f that varies regularly with index − − α such that G µ ( iy ) − iy = (cid:16) i − θ tan πα (cid:17) f ( y )(1 + o (1)) as y → ∞ . Unfortunately, although the proof in [12] is correct, the authors made a sign error when they wrotedown the statement. Thus, we have corrected the θ in the statement to a − θ , and we made thesame correction to the definition of ν α,θ . This result can be restated in terms of the K -transform andboolean convolution as follows. Proposition 7.11.
Let α ∈ (0 , ∪ (1 , . Then the following are equivalent:(1) µ ∈ C α,θ for α < or µ ∈ C α,θ for α > .(2) There exists a function g that varies regularly with index − α such that K µ ( iy ) = − (cid:16) i − θ tan πα (cid:17) g ( y )(1 + o (1)) as y → ∞ . (3) There exists a slowly varying function h such that c − /α · µ ⊎ c/h ( c /α ) → ν α,θ . Furthermore, in (2), we can take g ( y ) = − Im K µ ( iy ) .Remark . It follows immediately that ν α,θ ∈ C α,θ when α ∈ (0 ,
1) and ν α,θ ∈ C α,θ when α ∈ (1 , Proof. (1) ⇐⇒ (2). Observe that K µ ( iy ) = iy − F µ ( iy ) = iyF µ ( iy ) (cid:18) G µ ( iy ) − iy (cid:19) = − y (cid:18) G µ ( y ) − iy (cid:19) (1 + o (1)) . Of course, f ( y ) varies regularly with index − − α if and only if g ( y ) = y f ( y ) varies regularly withindex 1 − α . Thus, the previous proposition immediately implies the case where α < α > µ has mean zero. Now consider a general measure µ ∈ C α,θ for α >
1. Let c be the meanand let ν = ( − c ) + µ . Then K ν ( z ) = − c + K µ ( z + c ) and hence K µ ( iy ) = c − (cid:16) i − θ tan πα (cid:17) g ( y )(1 + o (1)) as y → ∞ , where g varies regularly with index 1 − α <
0. In particular, this implies that the second term on theright-hand side goes to zero. Hence, the mean c is uniquely recoverable from K µ . Furthermore, theright-hand side has the form − (cid:0) i − θ tan πα (cid:1) g ( y )(1 + o (1)) where g varies regularly with index 1 − α if and only if c = 0.(2) = ⇒ (3). Let g be as in (2), and write g ( y ) = y − α h ( y ) for some slowly varying function h . Because K -transforms are contained in the normal family Hol( H , − H ) (where the target space isthe closure in the Riemann sphere), to show c − /α · µ ⊎ c/h ( c /α ) → ν α,θ it suffices to prove pointwiseconvergence of the K -transforms on the imaginary axis. Note that K c − /α · µ ⊎ c/h ( c /α ) ( iy ) = c − /α h ( c /α ) K µ ( c /α iy )= − c − /α h ( c /α ) (cid:16) i − θ tan πα (cid:17) ( c /α y ) − α h ( c /α y )(1 + o c /α y (1))= − (cid:16) i − θ tan πα (cid:17) y − α h ( c /α y ) h ( c /α ) (1 + o c /α y (1)) , where the subscript on the o (1) term means that it vanishes as c /α y → ∞ . If y is fixed and c → ∞ ,then because g varies slowly, we obtain − (cid:16) i − θ tan πα (cid:17) y − α h ( c /α y ) h ( c /α ) (1 + o c /α y (1)) → − (cid:16) i − θ tan πα (cid:17) y − α = K ν α,θ ( iy ) . (3) = ⇒ (2). Suppose that (3) holds for some function h . Let g ( y ) = y − α h ( y ), so that g variesregularly with index 1 − α . Observe that K µ ( c /α i ) = c − /α h ( c /α ) K c − /α µ ⊎ c/h ( c /α ) ( i )= c − /α h ( c /α )( K ν α,θ ( i ) + o (1))= (cid:16) i − θ tan πα (cid:17) g ( c /α )(1 + o (1)) . where the error o (1) goes to zero as c → ∞ . Then we substitute c = y α and obtain (2). REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 23
For the final claim regarding g in (2), observe that − Im K µ ( iy ) ≥ − Im K µ ( iy ) = g ( y )(1 + o (1)). It is straightforward to check that this function varies regularly of index 1 − α . (The 1 + o (1)term in the original theorem statement is complex-valued, but the one used here is positive.) Thus,we can replace g ( y ) with K µ ( iy ) by absorbing g ( y ) /K µ ( iy ) into the 1 + o (1) term. (cid:3) We also need the following facts about regularly varying functions. They can be found in [17], butwe include an elementary proof here for the reader’s convenience.
Lemma 7.13. (1) If f varies regularly with index α and a > , then af varies regularly with index α .(2) If f varies regularly with index α and if β > , then f ( y ) β and f ( y β ) regularly with index αβ .(3) If f varies regularly with index α and if β ∈ R , then y β f ( y ) varies regularly with index α + β .(4) If f is bounded above and below on any compact set and varies regularly with index α = 0 , then lim y →∞ f ( y ) = ( ∞ , α > , , α < . (5) Let f : [ m, ∞ ) → [0 , ∞ ) be continuous and vary regularly with index α > . Let g ( y ) = inf { x ≥ m : f ( x ) ≥ y } . Then f ◦ g ( y ) = y for y > f ( m ) , and g varies regularly with index /α .Proof. Claims (1), (2), (3) are straightforward to check from the definition.To prove (4), suppose α > f ( y ) = y α g ( y ), where g varies slowly. Then there exists M > − α/ ≤ f (2 y ) f ( y ) ≤ α/ for y ≥ M. By hypothesis, f is bounded below by some δ on the set [ M, M ]. Any y ≥ M can be written as 2 n y ′ for y ′ ∈ [ M, M ] and n ≥
0, and then we have g ( y ) = g (2 n y ′ ) ≥ − nα/ g ( y ′ ) ≥ M α/ (2 n M ) − α/ δ ≥ M α/ δy − α/ . Therefore, y α/ g ( y ) has a positive lower bound for sufficiently large y , which implies that y α g ( y ) → ∞ .The case for α < /f .(5) Because f ( x ) → ∞ as x → ∞ , the infimum in the definition of g is well-defined. If y > f ( m ),then by continuity f must achieve the value y by the intermediate value theorem. Furthermore, f ( x ) < y for x in a neighborhood of m , and hence the infimum x of { x : f ( x ) ≥ y } must be strictlylarger than m . Then we have f ( x ) < y for x < x and there is a sequence of points converging to x from above that satisfy f ( x ) ≥ y , so by continuity f ( x ) = y , or f ( g ( y )) = y .Because f is bounded on any compact set, we must have g ( y ) → ∞ as y → ∞ . Given any t > ǫ >
0, since f varies regularly with index α , we havelim y →∞ f ((1 + ǫ ) /α c /α g ( y )(1 + ǫ ) cf ( g ( y )) = 1 . If y is large enough that the left-hand side is larger than 1 / (1 + ǫ ), then we obtain f ((1 + ǫ ) /α c /α g ( y )) ≥ cy. Thus, by definition of g , g ( cy ) ≤ (1 + ǫ ) /α c /α g ( y ) , so that lim sup n →∞ g ( cy ) c /α g ( y ) ≤ (1 + ǫ ) /α . Since ǫ was arbitrary, the lim sup is bounded above by 1. However, because the same thing holds with c replaced by 1 /c , we get lim inf n →∞ c − /α g ( cy ) g ( c (1 /c ) y ) ≥ . Therefore, g varies regularly with index 1 /α . (cid:3) We can conclude the proof as follows:
Proof of Theorem 7.9.
Let µ ∈ C α,θ if α ∈ (0 ,
1) and µ ∈ C α,θ if α ∈ (1 , h be as in Proposition7.11. By Lemma 7.13, the function c c/h ( c /α ) varies regularly with index 1. Let ψ ( t ) be thefunction associated to t/h ( t /α ) as in Lemma 7.13 (5), so that ψ ( t ) /h ( ψ ( t ) /α ) = t for sufficientlylarge t and ψ ( t ) varies regularly with index 1. Then let φ ( t ) = ψ ( t ) − /α . Then φ varies regularly withindex − /α , and φ ( t ) · µ ⊎ t = ψ ( t ) − /α · µ ⊎ ψ ( t ) /h ( ψ ( t ) /α ) → ν α,θ . In particular, for each
T ∈
Tree( N ) with n ( T ) >
1, we have φ ( n ( T ) k ) · µ ⊎ n ( T ) k → ν α,θ , and hence by Theorem 6.1, we have φ ( n ( T ) k ) · ⊞ T ◦ k ( µ ) → BP ( T , ν α,θ )for T ∈
Tree( N ) with n ( T ) >
1. That theorem also implies that the convergence is uniform over T . (cid:3) By appealing to Theorem 6.1, we did not have to check that C α,θ is closed under the operations µ ⊞ T ( µ ) or µ BP ( T , µ ) in order to prove Theorem 7.9. However, as one would intuitively hope,this is indeed the case. Proposition 7.14.
Let α ∈ (0 , and θ ∈ [ − , . Suppose µ ∈ C α,θ .(1) For t > , we have µ ⊎ t ∈ C α,θ .(2) For any T ∈
Tree( N ) with n ( T ) > , we have ⊞ T ( µ ) ∈ C α,θ (3) For T ∈
Tree( N ) with n ( T ) > , we have BP ( T , µ ) ∈ C α,θ .The same claims hold with C α,θ replaced by C α,θ for α ∈ (1 , .Proof. Let α ∈ (0 , µ ∈ C α,θ if andonly if there is a function f that varies regularly with index 1 such that c − /α µ ⊎ f ( c ) → ν α,θ as c → ∞ .In the remainder of the argument, assume µ ∈ C α,θ and let f be a function that varies regularlywith index 1 with c − /α µ ⊎ f ( c ) → ν α,θ .(1) If t >
0, then c − /α · ( µ ⊎ t ) ⊎ f ( c ) /t = c − /α · µ ⊎ f ( c ) → ν α,θ . The function f ( c ) /t also variesregularly with index 1, so µ ⊎ t ∈ C α,θ .(2) Let Φ N be as in Theorem 6.2. Then using Lemma 7.2, we have c − /α · ⊞ T ( µ ) ⊎ f ( c ) /n ( T ) = ⊞ T (( c − /α · µ ⊎ f ( c ) ) ⊎ /f ( c ) ) ⊎ f ( c ) /n ( T ) = Φ N (cid:16) T , N/f ( c ) , ( c − /α · µ ⊎ f ( c ) ) ⊎ /f ( c ) (cid:17) ⊎ N/n ( T ) . Of course, f ( c ) → ∞ as c → ∞ . Thus, by joint continuity of Φ N , we obtain thatlim c →∞ c − /α · ⊞ T ( µ ) ⊎ f ( c ) /n ( T ) = Φ N ( T , , ν α,θ ) ⊎ N/n ( T ) = ( ν ⊎ n ( T ) /Nα,θ ) ⊎ N/n ( T ) = ν α,θ . Thus, ⊞ T ( µ ) satisfies the desired condition with the function f /n ( T ).(3) Let n = n ( T ). As in the proof of Theorem 6.3, fix M ≥ ( N − / ( n − T k ∈ Tree(
M n k )be isomorphic to T ◦ k . Recall that BP ( T , µ ) = lim k →∞ ⊞ T ◦ k ( µ ⊎ /n k ) = lim k →∞ ⊞ T k ( µ ⊎ /n k )Then observe that c − /α · (cid:16) ⊞ T k (cid:16) µ ⊎ nk (cid:17)(cid:17) ⊎ f ( c ) = ⊞ T k (cid:16) ( c − /α · µ ⊎ f ( c ) ) f ( c ) nk (cid:17) ⊎ f ( c ) = Φ Mn k (cid:16) T k , M/f ( c ) , ( c − /α · µ ⊎ f ( c ) ) (cid:17) ⊎ M . REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 25
By Theorem 6.2, we havelim c →∞ Φ Mn k (cid:16) T k , M/f ( c ) , ( c − /α · µ ⊎ f ( c ) ) (cid:17) ⊎ M = Φ Mn k ( T k , , ν α,θ ) ⊎ M = (cid:18) ν ⊎ nkMnk α,θ (cid:19) ⊎ M = ν α,θ , and the rate of convergence is uniform for all k . Uniform convergence implies thatlim c →∞ lim k →∞ c − /α · (cid:16) ⊞ T k (cid:16) µ ⊎ nk (cid:17)(cid:17) ⊎ f ( c ) = lim k →∞ lim c →∞ c − /α · (cid:16) ⊞ T k (cid:16) µ ⊎ nk (cid:17)(cid:17) ⊎ f ( c ) , and hence lim c →∞ c − /α BP ( T , µ ) ⊎ f ( c ) = ν α,θ , so BP ( T , µ ) ∈ C α,θ .This concludes the proof for α ∈ (0 , α ∈ (1 ,
2) with C α,θ replaced by C α,θ . (cid:3) In the cases of α = 1 and α = 2, the tools which Bercovici and Pata used to prove the characteri-zation of C α,θ in terms of Cauchy transforms [12, Propositions 5.10 and 5.11] are not available in thesame form; specifically, [12, Proposition 5.8] does not handle the case α = 2 and the later parts ofthat proposition do not handle the case α = 1. To study the T -free convolution for the regions C ,θ and C requires either a much more delicate analysis or a different approach. We will be content hereto deduce limit theorems from the classical theory and Bercovici and Pata’s results. Proposition 7.15.
Let θ ∈ [ − , and let µ ∈ C ,θ . Then there exists a sequence of meaures ( µ j ) j ∈ N equivalent to µ such that, for all N , for all T ∈
Tree( N ) with n ( T ) , we have ⊞ T ◦ k ( µ n ( T ) k ) → BP ( T , ν ,θ ) , where for each N , the convergence is uniform over T . Proposition 7.16.
Let µ ∈ C with mean zero. Then there exists a sequence R j tending to infinitysuch that for all N , for all T ∈
Tree( N ) with n ( T ) , we have R − n ( T ) k · ⊞ T ◦ k ( µ ) → BP ( T , ν ) , where for each N , the convergence is uniform over T . To set the stage for the proof, we recall the results from [12] in more detail. The infinitely divisibledistributions for ∗ , ⊞ , and ⊎ are parametrized by a γ ∈ R and finite measure σ on R , and the infinitelydivisible distributions corresponding to ( γ, σ ) for the three convolutions are denoted respectively by ν γ,σ ∗ , ν γ,σ ⊞ , and ν γ,σ ⊎ . For a sequence of probability measures µ j and k j → ∞ , we have µ ∗ k j j → ν γ,σ ∗ ifand only if µ ⊞ k j j → ν γ,σ ⊞ if and only if µ ⊎ k j j → ν γ,σ ⊎ .It follows that for every N , BP ( T N, free , ν γ,σ ⊎ ) = lim k →∞ (( ν γ,σ ⊎ ) ⊎ /N k ) ⊞ N k = ν γ,σ ⊞ . Moreover, let Φ µ denote the Voiculescu transform Φ µ ( z ) = F − µ ( z ) − z , defined in a non-tangentialneighborhood of ∞ . The correspondence between the free and boolean cases is such that(7.1) Φ ν γ,σ ⊞ ( z ) = γ + Z R tzz − t dµ ( t ) = K ν γ,σ ⊎ ( z ) . It follows from [12, §
5] that the freely stable laws correspond precisely to the classically stable laws.However, these do not correspond to boolean stable laws in the na¨ıve sense. Rather, for a ∈ R , itfollows from (7.1) that BP ( T , free , δ a ⊎ µ ) = δ a ⊞ BP ( T , free , µ ) = a + BP ( T , free , µ ) , and thus stability in the boolean setting should be understood with respect to the shift operations µ δ a ⊎ µ for a ∈ R rather than µ a + µ . The laws ν α,θ in Proposition 7.8 above are the boolean stable laws with this modified notion of stability, and the freely stable distributions in [12, Proposition5.12] are exactly the distributions BP ( T N, free , ν α,θ ), where ν α,θ .Proposition 7.15 is now proved as follows: Let ρ α,θ be the classical stable distribution correspondingto the boolean infinitely divisible distribution ν α,θ . From classical results, if µ ∈ C ,θ , there aremeasures µ j equivalent to µ such that µ ∗ j j → ρ ,θ . Hence by [12, Theorem 6.3], we have µ ⊎ jj → ν α,θ .Then by Theorem 6.1, we have ⊞ T ◦ k ( µ n ( T ) k ) → BP ( T , ν ,θ ). The proof of Proposition 7.16 is thesame. 8. Open questions
The following questions around Theorems 6.1 and 7.9 remain unanswered.
Question 8.1.
Does the converse implication hold in Theorem 6.1? More precisely, let
T ∈
Tree( N )with n ( T ) >
2. If ⊞ T ◦ kℓ ( µ ℓ ) converges as ℓ → ∞ , then does µ ⊎ n ( T ) kℓ ℓ converge? A positive answer isknown for free independence by [12], but to our knowledge the question is open even in the case ofmonotone independence. Question 8.2.
Is the map BP ( T , · ) injective and is the inverse continuous? For compactly supportedmeasures, the inverse map was studied using combinatorial methods in [27, § Question 8.3.
What is the correct notion of stable law for T -free convolution? Do we get a classi-fication of such laws that is parallel to the classical case? Of course, this question is one of the mainmotivations for the previous two questions. Question 8.4.
Is there a limit theorem which allows us to bring the translation operation outsidethe convolution operations? That is, if µ ∈ C α,θ or C , then can we describe the asymptotic behaviorof some sequence µ k ∼ ⊞ T ◦ k ( µ )? Question 8.5.
Does Proposition 7.14 generalize to the α = 1 and α = 2 cases? What is the correctsubstitute for Proposition 7.11 in these cases?There are many interesting questions about the limiting distributions BP ( T , ν α,θ ) themselves.We know that ⊞ T ◦ k ( ν ⊎ n ( T ) − k α,θ ) → BP ( T , ν α,θ ), and in the case α ∈ (0 , ∪ (1 , ⊞ T ◦ k ( ν ⊎ n ( T ) − k α,θ ) = n ( T ) − /α · ⊞ T ◦ k ( ν α,θ ). Furthermore, the Stieltjes inversion formula says that un-der sufficient regularity conditions, the probability density of a measure µ can be recovered fromthe Cauchy transform by ρ ( x ) = lim ǫ → + − π G µ ( x + iy ). As an example, we considered the tree T = {∅ , , , , , , , } . To approximate the density for BP ( T , ν α,θ ), we computed − π Im G ⊞ T ◦ ( ν ⊎ − α,θ ) ( x + iǫ )for ǫ = 10 − and for values of x spaced at intervals of 0 .
1, and the results are shown in Figures 1and 2. Experimentally, replacing 6 by 7 or shrinking ǫ did not change the values much. However,because the size of the tree T ◦ k increases very quickly with k , this approximation scheme has highcomputational complexity and is thus impractical to evaluate for large k . Question 8.6.
Are there practical numerical error bounds for the convergence of ⊞ T ◦ k ( ν ⊎ n ( T ) − k α,θ ) → BP ( T , ν α,θ ? Similarly, what is the rate of convergence to ⊞ T ( µ , . . . , µ N ) of the approximations givenby truncation of T to finite trees? Are there better estimates for special classes of trees?Already for a single tree T = {∅ , , , , , , , } , we saw a variety of phenomena occur. For α ∈ (0 ,
1) there is a singularity at 0 in the boolean case, but in the free case the stable laws haveanalytic densities on their supports [12, Propositions A.1.2-A.1.4]. For this T , the presence or absenceof a singularity appears to depend on the value of α ∈ (0 , α ∈ (1 , REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 27 α = 1 . θ = 0 .
00 0 . . − − α = 1 . θ = 0 .
40 0 . . − − α = 1 . θ = 0 .
80 0 . . . . . − − α = 1 . θ = 0 .
00 0 . . − − α = 1 . θ = 0 .
40 0 . . − − − α = 1 . θ = 0 .
80 0 . . . . . − − − Figure 1.
Approximations of BP ( T , ν α,θ ) for T = {∅ , , , , , , , } and for( α, θ ) ∈ { . , . } × { . , . , . } . Question 8.7.
What can we say about the regularity of the limit distributions BP ( T , ν α,θ )? Do theyhave analytic densities? How does this vary with T , α , and θ ? In general, what can we say about theregularity of T -free convolutions of several measures? Under what conditions on T do the regularityresults from the free case [15, 14, 5, 8, 7] generalize?Another open question concerns the operator models for T -free convolution. In this paper, we fo-cused exclusively on the complex-analytic viewpoint for T -free convolutions, even though the originaldefinition of the convolution for compactly supported measures was in terms of addition of “indepen-dent” bounded self-adjoint operators [27]. Moreover, the free convolution of arbitrary measures on R α = 0 . θ = 0 .
00 0 . . − − α = 0 . θ = 0 .
40 0 . . − − α = 0 . θ = 0 .
80 0 . . − − α = 0 . θ = 0 .
00 0 . . − − α = 0 . θ = 0 .
40 0 . . − − α = 0 . θ = 0 .
80 0 . . − − Figure 2.
Approximations of BP ( T , ν α,θ ) for T = {∅ , , , , , , , } and for( α, θ ) ∈ { . , . } × { . , . , . } .can be expressed using the addition of freely independent unbounded self-adjoint operators, thanksthe theory of unbounded operators affiliated to a tracial von Neumann algebra [13]. Question 8.8.
Can the T -free convolution of arbitrary probability measures on R be formulated interms of addition T -free independent unbounded self-adjoint operators?Because arbitrary self-adjoint operators cannot necessarily be added, the challenge is to use theadditional structure of T -free independence (or perhaps of the T -free product Hilbert space) to showthat the sum actually makes sense. Again, we believe that the solution for finite-variance measures issignificantly easier than for the general case. References [1] Michael Anshelevich, Serban T. Belinschi, Maxime F’evrier, and Alexandru Nica. Convolution powers in theoperator-valued framework.
Trans. Am. Math. Soc. , 365:2063–2097, 2013.[2] Michael D. Anshelevich and John D. Williams. Limit theorems for monotonic convolution and the Chernoff productformula.
International Mathematics Research Notices , 2014(11):2990–3021, 2014.[3] Michael D. Anshelevich and John D. Williams. Operator-valued monotone convolution semigroups and an extensionof the Bercovici-Pata bijection.
Documenta Mathematica , 21:841–871, 2016.[4] Octavio Arizmendi, Miguel Ballesteros, and Francisco Torres-Ayala. Conditionally free reduced products of hilbertspaces. To appear in Studia Mathematica, 2019.
REE CONVOLUTION FOR PROBABILITY DISTRIBUTIONS WITH UNBOUNDED SUPPORT 29 [5] Serban T. Belinschi. The atoms of the free multiplicative convolution of two probability distributions.
IntegralEquations Operator Theory , 46(4):377–386, 2003.[6] Serban T. Belinschi. Complex analysis methods in non-commutative probability. Ph.D. thesis at University ofIndiana, 2006.[7] Serban T. Belinschi. A note on regularity for free convolutions.
Ann. Inst. H. Poincar´e Prob. , 42:635–648, 2006.[8] Serban T. Belinschi and Hari Bercovici. Atoms and regularity for measures in a partially defined free convolutionsemigroup.
Math. Z. , 248(4):665–674, 2004.[9] Serban T. Belinschi, Tobias Mai, and Roland Speicher. Analytic subordination theory of operator-valued freeadditive convolution and the solution of a general random matrix problem.
Journal f¨ur die reine und angewandteMathematik (Crelles Journal) , 03 2013.[10] Serban T. Belinschi, Mihai Popa, and Victor Vinnikov. On the operator-valued analogues of the semicircle, arcsineand Bernoulli laws.
Journal of Operator Theory , 70(1):239–258, 2013.[11] A. Ben Ghorbal and M. Sch¨urmann. Non-commutative notions of stochastic independence.
Math. Proc. Camb.Phil. Soc. , 133:531–561, 2002.[12] Hari Bercovici and Vittorino Pata. Stable laws and domains of attraction in free probability theory.
Ann. Math. ,149:1023–1060, 1999. With an appendix by Philippe Biane.[13] Hari Bercovici and Dan-Virgil Voiculescu. L`evy-Hincin type theorems for multiplicative and additive free convolu-tion.
Pac. J. Math. , 153:217–248, 1992.[14] Hari Bercovici and Dan-Virgil Voiculescu. Regularity questions for free convolution. In Hari Bercovici and Ciprian I.Foias, editors,
Nonselfadjoint Operator Algebras, Operator Theory, and Related Topics , volume 104 of
Oper. TheoryAdv. Appl. , pages 37–47. Birkh¨auser, Basel, 1998.[15] Philippe Biane. On the free convolution with a semi-circular distribution.
Indiana Univ. Math. J. , 46(3):705–718,1997.[16] Philippe Biane. Processes with free increments.
Mathematische Zeitschrift , 227(1):143–174, 1 1998.[17] N. H. Bingham, C. M. Goldie, and J. L. Teugels.
Regular Variation , volume 27 of
Encyclopedia of Mathematic andits Applications . Cambridge University Press, Cambridge, 1971.[18] Marek Bo˙zejko, Michael Leinert, and Roland Speicher. Convolution and limit theorems for conditionally free randomvariables.
Pacific J. Math. , 125(2):357–388, 1996.[19] Clifford J. Earle and Richard S. Hamilton. A fixed point theorem for holomorphic functions. In S. Smale andS. S. Chern, editors,
Global Analysis , volume 16 of
Proceedings of Symposia in Pure Mathematics , pages 61–65.American Mathematical Society, Providence, 1970.[20] Uwe Franz and Romuald Lenczewski. Limit theorems for the hierarchy of freeness.
Probab. Math. Stat. , 19:23–41,1999.[21] B. V. Gnedenko and A. N. Kolmogorov.
Limit Distributions for Sums of Independent Random Variables . Addison-Wesley Publ. Co., Cambridge, Mass., 1954.[22] Takahiro Hasebe. Monotone convolution and monotone infinite divisibility from complex analytic viewpoints.
Infin.Dimens. Anal. Quantum Probab. Relat. Top. , 13(1):111–131, 2010.[23] Takahiro Hasebe. Monotone convolution semigroups.
Studia Math. , 200:175–199, 2010.[24] Takahiro Hasebe. Conditionally monotone independence i: independence, additive convolutions and related convo-lutions.
Infinite Dimensional Analysis, Quantum Probability and Related Topics , 14(03):465–516, 2011.[25] Takahiro Hasebe and Hayato Saigo. On operator-valued monotone independence.
Nagoya Math. J. , 215:151–167,2014.[26] David Jekel. Operator-valued chordal loewner chains and non-commutative probability.
J. Func. Anal. ,278(10):108452, 2020.[27] David Jekel and Weihua Liu. An operad of non-commuative independences defined by trees.
Dissertationes Math-ematicae , 553:1–100, 2020.[28] Anna Dorota Krystek. Infinite divisibility for the conditionally free convolution.
Infin. Dim. Anal. Quantum Prob.and Relat. Top. , 10(04):499–522, 2007.[29] Anna Kula and Janusz Wysocza´nski. An example of a Boolean-free type central limit theorem.
Probab. Math.Statist. , 33:341–352, 2013.[30] Tom Leinster.
Higher Operads, Higher Categories , volume 298 of
London Mathemical Society Lectures Notes Series .Cambridge University Press, 2004.[31] Romuald Lenczewski. Unification of independence in quantum probability.
Infin. Dimens. Anal. Quantum Probab.Relat. Top. , 1:383–405, 1998.[32] Romuald Lenczewski. Decompositions of the free additive convolution.
Journal of Functional Analysis , 246(2):330–365, 2007.[33] Romuald Lenczewski. Operators related to subordination for free multiplicative convolutions.
Indiana Univ. Math.J. , 57:1055–1103, 2008.[34] Romuald Lenczewski. Conditionally monotone independence and the associated products of graphs.
Infin. Dimens.Anal. Quantum. Probab. Relat. Top. , 22(04):1950023, 2019.[35] Wojciech M lotkowski. λ -free probability. Infinite-dimensional Analysis, Quantum Probability, and Related Topics ,7:27–41, 2004. [36] Naofumi Muraki. Monotonic convolution and monotone L´evy-Hinˇcin formula. preprint, 2000.[37] Naofumi Muraki. Monotonic independence, monotonic central limit theorem, and monotonic law of small numbers.
Infinite Dimensional Analysis, Quantum Probability, and Related Topics , 04, 2001.[38] Naofumi Muraki. The five independences as natural products.
Infinite Dimensional Analysis, Quantum Probabilityand Related Topics , 6(3):337–371, 2003.[39] Naofumi Muraki. A simple proof of the classification theorem for positive natural products.
Probab. Math. Statist. ,33(2):315–326, 2013.[40] Mihai Popa and Victor Vinnikov. Non-commutative functions and the non-commutative L´evy-Hinˇcin formula.
Adv.Math. , 236:131–157, 2013.[41] Roland Speicher. On universal products. In Dan Voiculescu, editor,
Free Probability Theory , volume 12 of
FieldsInst. Commun. , pages 257–266. Amer. Math. Soc., 1997.[42] Roland Speicher. Combinatorial theory of the free product with amalgamation and operator-valued free probabilitytheory.
Mem. Amer. Math. Soc. , 132(627), 1998.[43] Roland Speicher and Reza Woroudi. Boolean convolution. In Dan Voiculescu, editor,
Free Probability Theory ,volume 12 of
Fields Inst. Commun. , pages 267–279. Amer. Math. Soc., 1997.[44] Roland Speicher and Janusz Wysocza´nski. Mixtures of classical and free independence.
Archiv der Mathematik ,107(4):445–453, 10 2016.[45] Dan-Virgil Voiculescu. Symmetries of some reduced free product C ∗ -algebras. In Huzihiro Araki, Calvin C. Moore,S¸erban-Valentin Stratila, and Dan-Virgil Voiculescu, editors, Operator Algebras and their Connections with Topol-ogy and Ergodic Theory , pages 556–588. Springer Berlin Heidelberg, Berlin, Heidelberg, 1985.[46] Dan-Virgil Voiculescu. Addition of certain non-commuting random variables.
Journal of Functional Analysis ,66(3):323–346, 1986.[47] Janusz Wysocza´nski. bm-independence and bm-central limit theorems associated with symmetric cones.
InfiniteDimensional Analysis, Quantum Probability and Related Topics , 13(03):461–488, 2010.[48] Max A. Zorn. Characterization of analytic functions in Banach spaces.
Ann. of Math. , 2, 1945.[49] Max A. Zorn. Gˆateaux differentiability and essential boundedness.
Duke Math. J. , 12:579–583, 1945.[50] Max A. Zorn. Derivatives and Fr´echet differentials.
Bull. Amer. Math. Soc. , 52:133–137, 1946.
Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095
Email address : [email protected] Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093
Email address : [email protected] Department of Mathematics, University of California, San Diego, La Jolla, CA, 92093
Email address ::