HALF INDEPENDENCE AND HALF CUMULANTS ARUP BOSE

in PROBABILITY

HALF INDEPENDENCE AND HALF CUMULANTS

ARUP BOSE¹

Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108.

email: [email protected] RAJAT SUBHRA HAZRA

Statistics and Mathematics Unit, Indian Statistical Institute, 203 B.T. Road, Kolkata 700108.

email: [email protected] KOUSHIK SAHA

Bidhannagar Government College, Sector I, Salt Lake, Kolkata 700064.

email: [email protected]

SubmittedNovember 2, 2010, accepted in final formApril 5,2011 AMS 2000 Subject classification: Primary 60B20; Secondary 46L53, 46L54, 60B10

Keywords: Cramer’s theorem, Central limit theorem, cumulant,C^∗probability space, free algebras, free independence, half commutativity, half independence, noncommutative probability spaces, reverse circulant matrix, Rayleigh distribution, semicircular law, symmetric partition.

Abstract

The notion of half independence arises in random matrices and quantum groups. This notion is available only for elements of a noncommutative probability space and assumes the existence of all moments. We relate half independence to a certain class of partitions and use it to define an appropriate cumulant generating function and a transform which is closely related to the charac- teristic function. This leads to a definition of half independent convolution of arbitrary probability measures which is compatible with the distribution of the sum of half independent elements of a noncommutative probability space. We also establish the central limit theorem for half indepen- dent convolution of measures with the limit being symmetrized Rayleigh. Cramer’s theorem is also established in this set up.

1 Introduction

Along with classical independence, another well known notion of independence isfree indepen- dencein the noncommutative set up. A third notion of independence in the noncommutative set up ishalf independence. This has been described in Banica, Curran and Speicher (2010)[2]and Bose, Hazra and Saha (2010)[7]. In Section 2, we provide a quick description of these three notions and two examples of half independence. The goal of this article is to study half independence in details.

1RESEARCH SUPPORTED BY J.C.BOSE FELLOWSHIP, GOVERNMENT OF INDIA. PART OF THE WORK DONE WHILE VISITING DEPARTMENT OF ECONOMICS, UNIVERSITY OF CINCINNATI IN OCTOBER 2010.

405

To motivate our results, let us recall the results from classical and free independence that are rele- vant to us. In classical independence, the two natural transforms, which are measure determining and convergence determining, are the characteristic functionφ(·)and the cumulant generating functionχ(·). For any probability measureµonR, these are defined as

φ_µ(t) = Z

R

e^{i t x}dµ(x) and χ_µ(t) =logφ_µ(t) (for t in a neighbourhood of 0).

If µ andν are two probability measures andµ∗ν is their independent (additive) convolution, then

χµ∗ν(t) =χµ(t) +χν(t). (1) Well known results for this convolution are the CLT and Cramer’s theorem.

Initially, free independence of noncommutative subalgebras or elements was defined via relations between moments. It is well known that noncrossing partitionsplay a crucial role in free inde- pendence. LaterCauchy transformand the R-transform(or Voiculescu transform) (see Nica and Speicher (2006)[10], Anderson et al. (2009)[1]) were defined. This helped to define free convo- lution of arbitrary probability measures. For instance theR-transform satisfies equation (1) when classical convolution is replaced by thefree convolution. Thefree CLTholds: thenfold free convo- lution of identical laws with finite second moment, when scaled byp

nand centered, converges to thesemicircle law. See Nica and Speicher (2006)[10]. However, Cramer’s theorem fails: free convolution of two laws may be semicircular without the individual laws being semicircular. See Bercovici and Voiculescu (1995)[5].

The existing notion of half independence makes sense only for elements of a noncommutative probability space via the behavior of their moments. It is known that this definition does not extend to half independence between subalgebras (see Speicher (1997)[11]) and no notion of half independence for arbitrary random variables is available in the literature.

The above article as well as the work of Bose, Hazra and Saha (2010)[7]suggests thatsymmetric partitionsplay a significant role in half independence. In Section 3 we use symmetric partitions to develop notions ofhalf cumulantsandhalf cumulant generating functionunder suitable restrictions on the growth of the moments. Then we relate the moments to the half cumulants via appropriate generating functions. This relationship helps us to go beyond the set up of noncommutative alge- bra and provides an analytical definition of half independent convolution of probability measures.

First, in Section 4 we develop the notion of an appropriate transform T_µ(·) for any symmetric (about zero) probability measureµ. This transform plays the same role in half independence as do the characteristic function in classical independence and Cauchy transform in free independence.

This transform T has a surprisingly simple description in terms of the characteristic function and is similar to the radial characteristic function defined in Kingman (1963)[8]. Let þdenote the product convolution. Then it turns out that

T_µ(t) =φµþρ(2t) for all t∈R whereρis the arcsine law with density

2 π

1

p1−α², 0< α <1.

T is both measure determining and convergence determining (see Lemma 2 below).

We define thehalf cumulant generating functionas,

H(t) =logT(t) (in an appropriate neighbourhood of 0).

Thehalf independent convolutionµν, of symmetric probability measuresµandνis defined as the (unique) probability measure which satisfies

T_µν(t) =T_µ(t)T_ν(t) for all t∈R. and hence the corresponding half cumulant generating functions satisfy

H_µν(t) =H_µ(t) +H_ν(t) (for t in a neighbourhood of 0).

This half convolution is quite similar to the convolution defined by Kingman (1963)[8] who ar- rived at it in a completely different context.

This definition of convolution is compatible with addition of half independent variables in a non- commutative probability space: ifaandbare half independent in someC^∗-probability space with lawsµandν, then the law of(a+b)exists and is given byµν.

In half independence Rayleigh law takes the place of Gaussian law in independence and the semi- circular law in free independence. Thesymmetrized Rayleigh law R_σ has the density

f(x) =|x|

σ²exp(−x²/σ²), −∞<x<∞ with moments

β2k+1=0 and β2k=σ^2kk! for all k≥0.

Ifσ=1, then it is known as standard symmetrized Rayleigh law and is denoted byR.

In Section 5, we establish Cramer’s theorem: µνis symmetrized Rayleigh, if and only if both µ andν are symmetrized Rayleigh. We also prove the half independent CLT: thenfold half in- dependent convolution of symmetric laws satisfying suitable moment condition, when scaled by pn, converges to the symmetrized Rayleigh distribution. An appropriate CLT in aC^∗-algebra also holds and is proved by computation of moments using symmetric partitions.

2 The three notions of independence

2.1 Preliminaries of noncommutative probability spaces

Let (A,τ) be a noncommutative∗-probability space whereA is a ∗−unital complex algebra (with unity 1) andτ:A →Cis a linear functional satisfyingτ(1) =1. At times we shall assume τis a state, that isτ(a)≥0 ifa≥0 (a≥0 meansa^∗=aand its spectrumsp(a)is nonnegative).

The elements of the algebra will be referred to as random variables and we shall concentrate only on self adjoint random variables. By the law of self adjointa∈ A, we mean the collection{m_k(a)}

where

m_k(a) =τ(a^k), k≥1 (2)

are themomentsofa. Supposeτis a state andais a self adjoint random variable. Then given any nand any collection of complex numbers{c_i, 0≤i≤n}

n

X

k=0 n

X

l=0

c_kτ(a^k+l)c¯_l=τ (

n

X

k=0

c_ka^k)(

n

X

l=0

c_la^l)^∗

!

≥0.

Then{m_k(a),k≥0} is a moment sequence, and there exists a measureµa on the real line such that

m_k(a) =τ(a^k) = Z

x^kdµa(x). (3)

µamay not be unique. A sufficient condition forµato be unique is Carleman’s condition

X[τ(a^2k)]^−1/(2k)=∞. (4)

In that case we also callµato be the law ofa.

When (A,τ) is a C^∗-algebra (that is, A is a ∗-algebra equipped with a normk · k such that ka bk ≤ kak kbkandkaa^∗k=kak²fora,b∈ A), the lawµaofaexists and is compactly supported.

More generally, suppose that the algebra is not necessarily C^∗ andτ is a state and for the self adjoint random variable a, τ(a^2k)≤ C^kk! for all k. Then {τ(a^2k)} satisfies condition (4), and henceµais the law ofabut it is not necessarily compactly supported.

For random variables{a_i}i∈J, theirjoint momentsare the collection{τ(a_i

1a_i

2. . .a_i

k), k≥1}, where eacha_i

j∈ {a_i}i∈J.

Random variables{a_i,n}i∈J∈(An,τn)are said toconverge in lawto{a_i}i∈J ∈(A⁰,τ⁰)if each joint moment of{a_i,n}i∈J converges to the corresponding joint moment of{a_i}i∈J.

2.2 Independence: free, classical, and half

Unital subalgebras{Ai}i∈J ⊂ A are calledfreely independentor simplyfreeif τ(a_j) =0, a_j∈ Ai_j and i_j6=i_j+1 for all j⇒τ(a₁· · ·a_n) =0.

Random variables (or elements of an algebra){a₁,a₂,· · · }are called free if the subalgebras gen- erated by them are free.

Unital subalgebras{Aj}j∈J⊂ A are said to beindependentif they commute and τ(a₁· · ·a_n) =τ(a₁)· · ·τ(a_n) for all a_j∈ Ai_j where k6=l⇒i_k6=i_l.

Two elementsaandbof any algebra are said to be independent if the two unital algebras gener- ated by them are independent.

Let{a_i}i∈J be noncommutative elements of(A,τ). We say that theyhalf commuteif a_ia_ja_k=a_ka_ja_i,

for alli,j,k∈J. Clearly, if{a_i}i∈J half commute thena²_i commutes witha_janda²_j for alli,j∈J. The random variablea=a_i1a_i2. . .a_in where eacha_ij∈ {a_i}i∈J, is said to bebalanced(with respect to{a_i}), if each random variablea_iappears same number of times in odd and even positions ofa.

Ifais not balanced, we say it isunbalanced. So ifnis odd thenais automatically unbalanced.

Definition 1. Half commuting elements{a_i}i∈J are said to be half independent if the following hold:

1. The variables{a_i²}i∈J are independent.

2. If a=a_i

1a_i

2· · ·a_i

nis unbalanced with respect to{a_i}i∈J, thenτ(a) =0.

This definition of half independent elements is equivalent to that given in Banica, Curran and Speicher (2009)[2]. Note that, from the second condition the odd moments of half independent elements are zero.

The three notions of independence are different from each other. See Bose, Hazra and Saha (2010)[7]for details. Here are two examples of half independence.

Example 1. (Banica, Curran and Speicher (2009)[2]) Let(Ω,B,µ)be a probability space and let {ψi}be a family of independent complex Gaussian random variables. Define a_i∈(M₂(L(µ)), E[tr(·)]) by

a_i=

– 0 ψi

ψi 0

™ .

wheretr(·)is normalized trace andψiis the complex conjugate ofψi. Then{a_i}are half independent.

The law of each a_iis a symmetrized Rayleigh distribution.

Example 2. (Bose, Hazra and Saha (2010)[7]) A reverse circulant matrix is an n×n symmetric matrix whose(i,j)-th entry is given by x_{(i+j) modn}. The sequence{x_i} is called the input sequence.

Let{RC_i,n}1≤i≤pbe an independent (across i) sequence of n×n reverse circulant matrices each with an independent input sequence with mean0, variance1and for all k≥1,

sup

n∈N

sup

1≤i≤p

sup

1≤m≤l≤n

E[|RC_i,n(m,l)|^k]≤c_k<∞.

Then{n^−1/2RC_i,n}1≤i≤p∈(M_n(L(µ)), E[tr(·)]), converges to half independent{a₁,a₂, . . . ,a_p}where a_iis as in Example 1 withE|ψi|²=1.

Incidentally, in Proposition 2.8 of Banica, Curran and Speicher (2010)[2]it was shown that if{x_i} are half independent elements in aC^∗ probability space, then there exists independent complex valued random variables{ψi}with E[ψⁿ_iψ^m] =0 form6=nandx_ihas the same distribution as

y_iwhere

y_i=

– 0 ψi

ψi 0

™ .

The above result reduces the study of half independence to the study of 2×2 matrices. However, we emphasize that this is true only in aC^∗probability space. There is no existing notion of half independence of arbitrary random variables or probability measures.

Remark 1. As pointed out above, there is no existing notion of half independence for random vari- ables affiliated to some von Neumann algebra. One may proceed along the lines of Bercovici and Voilculescu (1993)[4]to define half independence for self adjoint affiliated random variables. Given two symmetric measures µand ν, it would be interesting to get hold of two self adjoint affiliated random variables which are half independent and have lawsµandν. We provide some lead to this approach in Subsection 4.2.

In particular, given a sequence of measures with unbounded support, one does not know how to find half independent random variables with these distributions. We shall address this issue in Section 4.

3 Symmetric partitions and half independence

The proof of the above matricial limit (Example (2)) given in Bose, Hazra and Saha (2010)[7] suggests that there is a suitable class of partitions which is tied to the notion of half independence just as noncrossing partitions are tied to free independence. In this section, we show thatsymmet- ric partitionsmay be used to develop a suitable notion ofhalf cumulants. The correspondinghalf

cumulant generating functioncaptures the notion of half independence. This will serve as the pre- cursor of the general notion of half cumulants and half cumulant generating function for arbitrary symmetric probability measures to be developed in the next section.

Any setKof integers will be calledsymmetricif it has an equal number of odd and even integers.

Any partition πofK will be calledsymmetricif each partition block is symmetric. The set of all symmetric partitions of{1, 2, . . . , 2n}will be denoted byE(2n). Given a sequence{l_n:Aⁿ→C}

of maps and partitionπof{1, 2, . . . ,n}define l_π(a₁,a₂, . . . ,a_n) =Y

V∈π

l_|V|(a_j1, . . . ,a_j

|V|), (5)

whereV = (j₁,j₂,· · ·,j_|V|)is a block of the partitionπ.

Definition 2. Let(A,τ)be a noncommutative probability space. Let{a_i}i∈J be random variables in A and suppose thatτ(a_i

1· · ·a_i

k) =0when k is odd and i_i,· · ·,i_k ∈J . Then define half cumulants {r_k}of{a_i}i∈J, recursively by the following moment cumulant relation

τ(a_i1a_i2. . .a_ik) = X

π∈E(k)

r_π(a_i1,a_i2, . . . ,a_ik). (6) Observe that both sides of the moment cumulant relation above are equal to zero for odd values ofkand{r_n}are multilinear maps.

For any self adjointa∈(A,τ), thehalf cumulants of aare denoted by

r_n(a) =r_n(a,a, . . . ,a), n≥1. (7) Note that r_2n+1(a) =0 for alln≥0. Thehalf cumulant generating functionofais defined as the formal power series

H_a(t) =X

n≥1

(−1)ⁿr_2n(a)

(n!)² t²ⁿ. (8)

The reason for the extran! in the denominator shall be clear as we proceed.

Example 3. If the law of a is the symmetrized Rayleigh law R, then it is easy to see that

r₂(a) =1, r_2n(a) =0 (for n>1) and H_a(t) =−t². (9) Suppose{a_i}are half independent. Using (6) recursively, it is easy to see that for any evenk>1,

r_k(a_i

1,a_i

2, . . . ,a_i

k) =0 if a_i

1a_i

2. . .a_i

k is unbalanced. (10)

For example, let{a₁,a₂} ∈(A,τ), be half independent. Then

r₂(a₁,a₂) =τ(a₁a₂) =0 and r₂(a₂,a₁) =τ(a₂a₁) =0.

Again using (6),

r₄(a₁,a₁,a₁,a₂) =τ(a₁a₁a₁a₂)−r₂(a₁,a₁)r₂(a₁,a₂)−r₂(a₁,a₁)r₂(a₁,a₂) =0, r₄(a₁,a₂,a₁,a₂) =τ(a₁a₂a₁a₂)−r₂(a₁,a₂)r₂(a₁,a₂)−r₂(a₁,a₂)r₂(a₁,a₂) =0.

Classical independence and freeness are characterized by classical cumulants and free cumulants respectively (see Theorem 5.3.15 of Anderson et. al. (2009)[1] or Theorem 11.16 of Nica and Speicher (2006)[10]for the free cumulant results). The corresponding characterization for half independent random variables is the following:

Theorem 1. Let{a_i}_1≤i≤lbe a sequence of self adjoint half commuting random variables and suppose for all1≤j≤l, a_joccurs k_jtimes in a=a_i

1a_i

2. . .a_i

2n.

(i) Suppose{a_i} are half independent and a is balanced with respect to{a_i}_1≤i≤l. If k_i,k_j≥ 2for some1≤i,j≤l, then r_2n(a_i1,a_i2, . . . ,a_i2n) =0.

(ii) If τ(a_i1a_i2. . .a_i

k) =0whenever k is odd and r_2n(a_i1,a_i2, . . . ,a_i2n) = 0whenever k_i,k_j ≥1 for some1≤i,j≤l, then{a_i}_1≤i≤l are half independent.

Proof of the above theorem is a little long and we postpone it to Section 6.

An important consequence of Theorem 1 is the following corollary which is analogous to the additivity of cumulant generating function andR-transform for classical independence and free independence respectively.

Corollary 1. If a and b are self adjoint, half commuting and half independent then r_2n(a+b) =r_2n(a) +r_2n(b) for all n≥1, and H_a+b(t) =H_a(t) +H_b(t), where the second equality holds between formal power series.

Proof. Note thatr₂(a,b) =τ(a b) =0. Hence the result is true forn=2 since r₂(a+b) = r₂(a+b,a+b)

= r₂(a,a+b) +r₂(b,a+b)

= r₂(a,a) +r₂(a,b) +r₂(b,a) +r₂(b,b)

= r₂(a) +r₂(b). Forn>2,

r_2n(a+b) =r_2n(a+b, ..,a+b)

=r_2n(a,a+b, ...,a+b) +r_2n(b,a+b, ...,a+b)

=r_2n(a,a...,a) +r_2n(b,b, ...,b) + X

(a₁,a₂,...a_2n):a_i∈{a,b},a_i6=a∀i

r_2n(a₁,a₂, . . . ,a_2n). Now if(a₁,a₂, . . . ,a_2n)is unbalanced then by (10), r_2n(a₁,a₂, . . . ,a_2n) =0. If(a₁,a₂, . . . ,a_2n)is balanced buta,bboth appear in the tuple, then by Theorem 1 (i),r_2n(a₁,a₂, . . . ,a_2n) =0. So only the first two terms survive in the last expression and hencer_2n(a+b) =r_2n(a) +r_2n(b).

We now proceed to express the relation between moments and half cumulants of a random vari- able in terms of appropriate generating functions. This will leads us to the appropriate generating functions for arbitrary probability measures in the next section. Recall that from Definition 2,

m_2n(a) = X

π∈E(2n)

r_π(a). Theorem 2.

m_2n(a) =

n−1X

s=0

n s

n−1 s

r_2(n−s)(a)m_2s(a).

Proof. We adapt the proof of Proposition 7.7 of Banica, Curran and Speicher (2009)[3] for the recursion formula for cardinality ofE(2n). Fix anyπ∈E(2n). Then each partition block has same number of odd and even members. Now place the sequence of odd numbers in one row and place the even members in the next row. In this way, E(2n)is viewed as the set of partitions between the upper and lower rows ofnpoints such that each partition block has same number of upper and lower members. Now to form any such partition, we perform the following steps:

• First pick a numbers∈ {1, 2, . . . ,n}. This shall be the total number of odd elements in the partition block containing 1.

• Connect 1 in the upper row to someother(s−1)points in the upper row. So we have chosen all theoddmembers of the partition block that contains 1.

• Now choosespoints in the lower row and connect them to the already connecteds points in the upper row, thus completing the symmetric partition block of size 2scontaining 1.

• Now partition the remaining(n−s)odd points of the upper row and the remaining(n−s) even points in the lower row in any symmetric way to essentially partition a set of size 2(n−s)in a symmetric way.

The number of such partitions is clearly ⁿ⁻¹_s

−1

n s

#E_2(n−s). Now observe that for any fixeds, the total contribution tom_2n(a)from all these partitions equalsr_2s(a)m_2(n−s)(a). Hence

m_2n(a) =

n

X

s=1

n−1 s−1

n s

r_2s(a)m_2(n−s)(a)

=

n−1X

s=0

n−1 s

n s

r_2(n−s)(a)m_2s(a).

We now convert the above relation to a relation between appropriate generating functions of {m_2k}and{r_2k}. We drop the argumentain these expressions for ease of notation. Consider the following two formal power series:

T˜(t) = X∞

n=0

m_2nt²ⁿ

(n!)² and ˜H(t) = X∞

n=0

r_2nt²ⁿ

(n!)². (11)

Their formal derivatives are:

T˜⁰(t) =2 X∞

n=1

m_2nt²ⁿ⁻¹

n!(n−1)! and ˜H⁰(t) =2 X∞

n=1

r_2nt²ⁿ⁻¹ n!(n−1)!.

Corollary 2. The above formal power series are related by the following relation,T˜⁰(t) =T˜(t)H˜⁰(t) where T˜(0) =1andH˜(0) =1.When considered as appropriate functions, this differential equation has a solutionT˜(t) =exp(H˜(t)−1).Hence, formallyH˜(t) =1+log ˜T(t).

Proof. Using Theorem 2, T˜⁰(t) =2

X∞

n=1

m_2nt²ⁿ⁻¹ n!(n−1)!=2

X∞

n=1 n−1

X

s=0 n s

(n−1) s

(n−s)!(n−s−1)!r_2(n−s)m_2st²ⁿ⁻¹.

=2 X∞

n=1 n−1

X

s=0

1

(s!)²n!(n−1)!r_2(n−s)m_2st²ⁿ⁻¹=2 X∞

s=0

X∞

n=s+1

r_2(n−s)t^2(n−s)−1 (n−s)!(n−s−1)!

m_2st^2s (s!)²

=2 X∞

s=0

X∞

l=1

r_2lt^2l−1 l!(l−1)!

m_2st^2s

(s!)² =T˜(t)H˜⁰(t). Rest of the claims now follow easily.

4 T -transform

In this section we introduce a new transform, which we call the T-transform, appropriate for the study of half independence. We show that the T-transform is very closely related to the characteristic function.

4.1 T -transform for general probability measures

The finiteness of the two power series in (11) impose growth conditions on the moments, thereby restricting the class of variables (or measures) for which the transforms are defined. We now remedy this.

When a ∈(A,τ) is a self adjoint random variable with a compactly supported lawµa, using Fubini’s theorem,

T˜(t) = X∞

n=0

m_2n(a)t²ⁿ (n!)² =

Z

R

X∞

n=0

(x t)²ⁿ

(n!)² dµa(x) = Z

R

I₀(2x t)dµa(x) (12) where

I₀(t) = X∞

n=0

(t/2)²ⁿ (n!)² = 1

π Zπ

0

cosh(tcosθ)dθ

is the modified Bessel function of order 0 (see Section 3.7 of Watson (1995)[12]).

Supposeµais symmetric about 0. Letφµadenote the characteristic function ofµa. Then we know thatφµa(t)is real. Using this and (12),

T˜(i t) = 1 π

Z

R

Zπ

0

cosh(2x i tcosθ)dθdµa(x)

= 1

π Zπ

0

Z

R

cos(2x tcosθ)dµa(x)dθ

= 1

π Zπ

0

φµa(2tcosθ)dθ (since µa is symmetric about 0). We have thus arrived at the following definition:

Definition 3. For any lawµwhich is symmetric about zero (but does not necessarily have compact support), define the transform,

T_µ(t) =T˜(i t) = 1 π

Zπ

0

φ_µ(2tcosθ)dθ, t∈R.

We shall also useT_ato denote the transformT_µa. Ifµis compactly supported or more generally if m_2n=R

Rx²ⁿdµ(x)≤Cⁿn! for alln, then it easily follows from the above discussions that T_µ(t) =

X∞

n=0

m_2n(−1)ⁿt²ⁿ

(n!)² for all t∈R. (13)

Example 4. Ifµis symmetrized Rayleigh R_σ, then m_2n=σ²ⁿn!for all n and hence T_µ(t) =

X∞

n=0

m_2n(−1)ⁿt²ⁿ (n!)² =

X∞

n=0

(−1)ⁿ(σt)²ⁿ

n! =exp(−σ²t²), t∈R.

We now connect T andφthrough product convolution. For probability measuresµonRandρ on(0,∞), we use the product convolutionþwhere

(µþρ)(B) = Z∞

0

µ(x⁻¹B)ρ(d x) = (µ⊗ρ)f⁻¹(B) with f(x,y) =x y, (14)

for any Borel setB. This gives a probability measure onRsinceµ(x⁻¹R) =1 for everyx6=0. We skip the proof of the following Lemma.

Lemma 1. Letρbe the arcsine law with density ²

π

p1

1−α² for0< α <1. Ifµis a probability measure onRwhich is symmetric around0, then

T_µ(t) =φ_µþρ(2t).

Remark 2. Motivated by random walks with spherical symmetry, Kingman (1963)[8](see Section 4) defined the radial characteristic function ⁿψX(t) (in n dimensions) of a nonnegative random variable X with lawµ. Interestingly the final expression for T -transform is related to this:

T_µ(t) =ⁿψX(2t) for n=2.

Kingman showed that the radial characteristic function is measure determining and convergence de- termining for nonnegative random variables. Although we are dealing with measures symmetric about 0, we can use Kingman’s result to derive similar properties for T -transform.

The next Lemma follows easily from Lemma 2 and Theorem 2 of Kingman (1963)[8]respectively or it can be proved directly using the theory of Mellin transform and weak convergence.

Lemma 2. (i) Letµandνbe two symmetric (about zero) probability measures. Then T_µ(t) =T_ν(t) for all t∈Rif and only ifµ=ν.

(ii) Suppose{µn}andµare symmetric probability measures. Thenµnconverges toµweakly, if and only if T_µ

n(t)converges to T_µ(t)for all t∈R.

SinceT_µ(t) =φµþρ(t)andφis continuous at 0, the logarithm ofT is well defined in a neighbour- hood of 0. This leads us to the following definition of half cumulant generating function which is more general than the one given in (8).

Definition 4. The half cumulant generating function of any symmetric probability measure µ is defined as

H_µ(t) =logT_µ(t) (in an appropriate neighborhood of zero).

The half cumulant generating function H_a of any a in (A,τ) is H_µa whenever µa exists and is symmetric.

Example 5. Ifµis symmetrized Rayleigh R_σ, (see Example 4), then T_µ(t) =exp(−σ²t²)is non-zero for all t∈R. So

H_µ(t) =−σ²t² for all t∈R.

Conversely, if H_µ(t) =−σ²t², thenµis the symmetrized Rayleigh law R_σ.

4.2 Half convolution of two symmetric measures

As mentioned before, there is no existing notion of half independence for unbounded operators.

Here we do not give a definition for half independence for such operators but we define a suitable notion of half convolution for arbitrary probability measures with the help of the above transforms.

We proceed as follows: suppose(A,τ)is a C^∗-probability space anda,b ∈(A,τ)are two self adjoint, half independent random variables with compactly supported measures µa andµb on R respectively. Then obviously, µa and µb are symmetric and in a neighbourhood of zero the following relations hold.

H_a+b(t) =H_a(t) +H_b(t) and T_a+b(t) =T_a(t)T_b(t). (15) In this case the measureµa+bis the measure corresponding to the random variable(a+b)∈(A,τ) and is compactly supported and symmetric. Note that from the second relation of (15) we have

φ_µa+bþρ(t) =φ_{(µaþρ)∗(µbþρ)}(t)

in a neighbourhood of zero. As the measures above are compactly supported the relation holds for allt∈Rand henceT_a+b(t) =T_a(t)T_b(t)for allt∈R.

We now define the half convolution (notation: ) of two arbitrary symmetric measuresµ and ν. First assume that µandν are compactly supported measures. Now using similar argument as given in Proposition 2.8 of Banica, Curran and Speicher (2010)[2], we can construct two half independent random variables y₁,y₂as follows:

LetX₁,X₂be two independent random variables such thatX₁andX₂have lawµandνrespectively.

LetU₁,U₂be two independent Haar unitary random variables which are independent fromX₁,X₂ and letξi=U_iX_i. Thenξ1,ξ2are independent and

E[ξⁿiξ¯^mi ] =E[Xⁿ_i^+m]E[U_iⁿU¯_i^m] = δnmE[X_i^n+m]

=

E[X²ⁿ_i ] if n=m

0 if n6=m. (16)

Now define the variables y₁,y₂as

y_i=

0 ξi

ξ¯i 0

.

Then it is easy to check using (16) thaty₁,y₂are half independent and have lawsµ,νrespectively.

Then from Lemma 1

T_y1(t) =T_µ(t) =φµþρ(2t) and T_y2(t) =T_ν(t) =φνþρ(2t). Consider the productT_µ(t)T_ν(t). Then due to relation (15),

T_µ(t)T_ν(t) =T_y

1(t)T_y

2(t) =T_y

1+y₂(t) =T_β(t) =φβþρ(2t),

whereβis a symmetric measure corresponding to random variable(y₁+y₂)and this is unique by Lemma 2 (i). We defineβas half convolution ofµandν, that is,

µν:=β.

Now supposeµandν are two arbitrary symmetric measure. Define forn∈N, the two measures µnandνnby

µn(B) =µ(B∩[−n,n]) and νn(B) =ν(B∩[−n,n]), for any Borel set B⊆R.

Thenµnandνnare compactly supported symmetric measure onRandµn,νnconverges weakly toµ,νrespectively. Then as before we have measuresβnsuch that

T_µn(t)T_νn(t) =φ_µnþρ(2t)φ_νnþρ(2t) =φ_βnþρ(2t).

Sinceµn,νnconverges weakly toµ,νrespectively,φ_µnþρ(2t)φ_νnþρ(2t)converges and henceφ_βnþρ(2t) converges for allt. Henceβnþρconverges weakly to some measure onR. It easily follows that {βn}is tight andβnþρconverges weakly toβþρfor some unique measureβ. Uniqueness ofβ follows from Lemma 2(i) and we haveT_µ(t)T_ν(t) =T_β(t)for allt ∈R. Hence we arrive at the following definition.

Definition 5. For any two symmetric probability measuresµandν, their half independent convolu- tion is defined as the unique measureβsuch that T_µ(t)T_ν(t) =T_β(t)for all t∈R.We write

µν:=β.

Note thatis associative and commutative on the space of symmetric measures.

Remark 3. This convolution is quite similar to the notion of radial sum in n dimensions of two independent nonnegative random variables X,Y defined in Section 3 of Kingman (1963)[8].

5 Some properties of half independent convolution

In this section, we establish the CLT and Cramer’s theorem for half independent convolutions. For clarity, we present both, the algebraic version and the measure version in each case. First the Cramer’s theorem.

Theorem 3. (i) Let a and b be self adjoint and half independent random variables of a∗-probability space(A,τ)whereτ is a state. Supposeµ^a+bp

2

is R_σ. Then µa andµb exist, and are symmetrized Rayleigh.

(ii) Ifµandνare symmetric probability measures such thatµνis symmetrized Rayleigh, then both µandνare symmetrized Rayleigh.

Proof. (i) Without loss, assume thatσ=1. Then τ(((a+b)/p

2)^2k) =k!.

From this, and half independence, it is easy to see that

τ(a^2k+1) =0 and τ(a^2k)≤2^kk!. (17) Thus there exists unique symmetric probability measuresµaandµbsuch that

τ(a^k) = Z

x^kdµa(x) and τ(b^k) = Z

x^kdµb(x).

It easily follows that the characteristic functions ofµaandµbare real and hence they are symmet- ric about origin. Hence for allt,

T_p^a

2(t)Tpb 2(t)

=exp(−t²).

So we have

€φµaþρ(2t)Š €

φµbþρ(2t)Š

=exp(−t²).

Hence, by the classical Cramer’s theorem it follows that each of the product on the left side must be the characteristic function of a normal distribution. So T_p^a

2(t) =exp(−_σ^t22) for some σ. As a consequence, by the uniqueness Lemma 2, a is symmetrized Rayleigh R^p_2/σ. Likewise, b is symmetrized RayleighR^p_2/σ0.

The proof of the second part follows easily from the above arguments.

The next result is a CLT for half independent random variables in a noncommutative∗-probability space(A,τ). It is relevant to recall that the notion of half independence holds in any such space.

However, the corresponding probability laws may not exist and hence the developments of the previous section do not apply. As a consequence, the proof of this theorem is based solely on Definition 1 and counting via symmetric partitions.

Theorem 4. Let{x_i}be a sequence of self adjoint half independent random variables in(A,τ)with τ(x_i) =0andτ(x²_i) =1andsup_iτ(x^k_i)<∞for every k. If S_n= x₁+x₂+· · ·+x_n, then S_n/p

n converges in law to the standard symmetrized Rayleigh distribution R.

Proof. To show the required convergence it is enough to show that, τ(S^2k_n )

n^k →k! and τ(S^2k_n ⁺¹) n^k+1/2 →0.

By applying half independence and unbalancedness of each term in the expansion of(x₁+· · ·+ x_n)^2k+1, it immediately follows thatτ((x₁+· · ·+x_n)^2k+1) =0. It is thus enough to consider the even moments.

Consider anyq= x_i1· · ·x_i2k in the expansion of(x₁+· · ·+x_n)^2k which is unbalanced. Then by half independenceτ(q) =0. Thus we are left with only balanced monomialsq. We divide such monomials into two sets:

M₁={x_i

1· · ·x_i

2k;i_j∈ {1, 2,· · ·,n}balanced and every random variable appears exactly twice}, M₂={x_i

1· · ·x_i

2k;i_j∈ {1, 2,· · ·,n}balanced and at least one random variable appears more than twice}. Observe thatM₂has at most(k−1)many distinct random variables and hence

#M₂≤C_kn(n−1)· · ·(n−k+2),

for some constantC_kthat depends only onk. Now since all moments are finite, 1

n^k X

M₂

τ(x_i1· · ·x_i2k)≤C_k⁰n(n−1)· · ·(n−k+2)

n^k →0 as n→ ∞.

Now consider M₁. Pick a fixed set of k variables from {x₁, . . . ,x_n}. Then there are exactly k!

ways of forming a monomial of length 2kof thesekvariables where each of the variables appears exactly once in an odd position and once in an even position. Hence

#M₁=k!×n(n−1)· · ·(n−k+1). Therefore asn→ ∞

1 n^k

X

M₁

τ(x_i

1· · ·x_i

2k) =#M₁

n^k τ(x₁²· · ·x_k²) = k!×n(n−1)· · ·(n−k+1)

n^k →k!.

Hence the result follows.

The CLT for half independent convolution of measures is stated next. Its proof is easy and is in spirit similar to the proof of Theorem 6 in Kingman (1963)[8].

Theorem 5. Let{µn}be a sequence of symmetric probability measures with variance1.Suppose for everyε >0,

1 n

n

X

k=1

Z

{x:|x|≥εp n}

x²dµk(x)→0 as n→ ∞ (Lindeberg’s condition). (18) Then

δn:=D_p¹

nµ1D_p¹

nµ2 · · · D_p¹

nµn

converges weakly to R where D_cµi(B) =µi(c⁻¹B)for any Borel set B inR. Proof. By definition of half convolution

T_δ

n(t) = T_D1

pnµ1(t)T_D1

pnµ2(t)· · ·T_D1

pnµn(t)

= φDp1

nµ1þρ(2t)φDp1

nµ2þρ(2t)· · ·φDp1

nµnþρ(2t), whereρis as defined in Lemma 1. Note that for anyε >0

1 n

n

X

k=1

Z

{x:|x|≥εp n}

x²d(µkþρ)(x) = 1 n

n

X

k=1

2 π

Z1

0

Z

{x:|xα|≥εp n}

x²α²

p1−α²µk(x)dα

= 2

π Z1

0

α² p1−α²



 1 n

n

X

k=1

Z

{x:|xα|≥εp n}

x²dµk(x)



dα(19). Now asn→ ∞, last expression in (19) goes to 0 by dominated convergence theorem and condition (18). Hence the sequence of measure{µnþρ}satisfies Lindeberg’s condition. Also

Z1

0

x²d(µkþρ)(x) = 2 π

Z1

0

1 p1−α²

Z

R

x²α²dµk(x)dα= 1 2. Hence by classical central limit theorem (see Billingsley (1995)[6]) asn→ ∞,

T_δ

n(t)→φN

0, 12(t) =exp(−t²) whereN_0,¹

2 is the Gaussian measure with mean zero and variance 1/2. We knowT_R(t) =exp(−t²). Henceδnconverges weakly to the standard (symmetrized) Rayleigh measureR.

6 Proof of Theorem 1

Proof. (i) For convenience, denote the trivial partition of{1, 2, . . . , 2n}with one single block by I_2n. We prove the first part of the theorem through induction ond (which equals the number of distinct random variables ina=a_i1a_i2. . .a_i2n) andn. For each fixed values ofdwe use induction onn. We use notationD₁for induction ondandD₂for induction onn.