On Boolean selfdecomposable distributions

This paper introduces the class of selfdecomposable distributions concerning Boolean convolution. A general regularity property of Boolean selfdecomposable distributions is established; in particular the number of atoms is at most two and the singular continuous part is zero. We then analyze how shifting probability measures changes Boolean selfdecomposability. Several examples are presented to supplement the above results. Finally, we prove that the standard normal distribution $N(0,1)$ is Boolean selfdecomposable but the shifted one $N(m,1)$ is not for sufficiently large $|m|$.


Introduction
In non-commutative probability theory, random variables are defined as elements in some * -algebra.A remarkable aspect of this theory is that various notions of independence exist for those random variables.
From a certain viewpoint, notions of independence are classified into five ones: tensor, free, Boolean, monotone and anti-monotone independences (see [18]).Further, each notion of independence associates the convolution of (Borel) probability measures on R that is defined to be the distribution of the sum of two independent (self-adjoint) random variables having prescribed distributions.
Limit theorems have been central subjects in both commutative (classical) and non-commutative probability theories.Among all, Khintchine introduced the class L of limit laws of certain independent triangular arrays.More precisely, a probability measure µ on R belongs to L if there exist a sequence of independent R-valued random variables {X n } n≥1 and sequences of deterministic numbers {a n } n≥0 ⊂ R and {b n } n≥1 ⊂ (0, ∞) such that • the family {b n X k } n≥k≥1 forms an infinitesimal triangular array, i.e.
• the law of the sequence of random variables converges weakly to µ.
When {X n } n≥1 is furthermore identically distributed, the limit distribution µ, if exists, is known to be a stable distribution.Thus the class L contains the class of stable distributions as a subset.In 1937, Lévy characterized the class L by the following property (see e.g.[11, Theorem 1, Section 29]): A probability measure µ on R is in L, if and only if µ is selfdecomposable, i.e., for any c ∈ (0, 1), there exists a probability measure µ c on R, such that µ = D c (µ) * µ c , where D c (µ) is the push-forward of µ by the mapping x → cx, and * denotes the classical convolution (see e.g.[20]).
In non-commutative probability theory, an analogous limit theorem can be formulated for each notion of independence.Bercovici and Pata [6], Chistyakov and Goetze [9] and Wang [22] proved a parallelism between classical Khintchine's limit theorem above and its non-commutative versions (for further details see Subsection 2.3 below).Correspondingly, notions of selfdecomposable distributions with respect to other convolutions are also defined (see Definition 3.1 below).In particular, the two classes of freely and monotonically selfdecomposable distributions were investigated in details (see e.g.[2,13,14] and [10], respectively), where analogy and disanalogy with the classical class L are discussed.By contrast, little had been done on Boolean selfdecomposable distributions.Although our definition of Boolean selfdecomposable distributions is so natural for specialists, there had been no formal definition in the literature to the authors' best knowledge.
The purpose of this paper is to study the class of Boolean selfdecomposable distributions.Results will be presented along the following lines.In Section 2, we introduce some concepts and some preliminary results that are used in the remainder of the paper.In Section 3, we establish a general regularity result on Boolean selfdecomposable distributions.Especially, we demonstrate that the Boolean selfdecompsable distributions have at most two atoms and do not have singular continuous part.We then investigate how Boolean selfdecomposability changes under shifts.It turns out that Boolean selfdecomposability is typically broken when the distribution is shifted with a sufficiently large positive or negative number.Furthermore, we observe some distributional properties of Boolean selfdecomposable distributions through several examples.In Section 4, we study Boolean selfdecomposability for normal distributions.The results in this section are motivated by the fact that every normal distribution is both classically and freely selfdecomposable (see e.g.[20] and [13], respectively).Our conclusion is that the standard normal distribution N(0, 1) is Boolean selfdecomposable too; however, the shifted one N(m, 1) is not Boolean selfdecomposable when |m| is sufficiently large as a consequence of the aforementioned general result in Section 3. Simulations suggest that |m| ≃ 3.09 is the approximate threshold.

Preliminaries
In the first of this section, we shall introduce analytic tools for understanding Boolean and free additive convolutions.After that we summarize a transfer principle for limit theorems for convolutions, which induces a bijection between different kinds of infinitely divisible distributions.

Boolean convolution and analytic tools
The Boolean convolution of probability measures on R was defined in [19].We set P (R) to be the set of all Borel probability measures on R, C + the complex upper half-plane and C − the complex lower halfplane.For µ ∈ P (R), we define the Cauchy transform G µ : C + → C − and the F-transform F µ : C + → C + as follows: The self-energy function K µ of µ is defined by The self-energy functions are characterized in the following way (see [19,Proposition 3.1] for further details).
Proposition 2.1.Let K be an analytic function on C + .The following assertions are equivalent: (1) There exists µ ∈ P (R) such that K = K µ .
(2) There exist b ∈ R and a finite positive measure τ on R, such that K has the form If either of (1) or (2) holds, then the pair (b, τ) is uniquely determined by µ (from formulas (2.2)-(2.3)below).
We call such a pair (b, τ) the Boolean generating pair for µ.By Lemma 2.1, there is a one-to-one correspondence between the set P (R) and the set of all Boolean generating pairs.Hence, we denote by µ (b,τ) ⊎ the probability measure µ with a Boolean generating pair (b, τ).The Boolean generating pair for µ (b,τ) ⊎ can be computed from the following formulas: for all α, β ∈ R that are continuity points of τ.The latter formula is referred to as the Stieltjes inversion formula.
For µ, ν ∈ P (R), their Boolean convolution µ ⊎ ν is the probability measure on R satisfying The η-transform of µ ∈ P (R) is defined by It is obvious to see that η µ (z) = zK µ (1/z), and therefore A probability measure µ is said to be Boolean infinitely divisible if for each n ∈ N there exists . For each µ ∈ P (R) with a Boolean generating pair (b, τ), let us set µ n as the probability measure with Boolean generating pair (b/n, τ/n) for each n ∈ N. Then µ is the n-fold Boolean convolution of µ n .Therefore every probability measure on R is Boolean infinitely divisible.
For a probability measure µ on R with a Boolean generating pair (b, τ), let us set (2.5) The triplet (a, ν, γ) thus defined fulfills (T) a ≥ 0, γ ∈ R and ν is a positive measure on R such that ν({0}) = 0 and R (1 Moreover, the set of triplets (a, ν, γ) satisfying (T) is in bijection with the set of pairs (b, τ) of a real number b and a finite positive measure τ on R. In terms of this bijection, formula (2.1) has the following equivalent form The real number a ≥ 0 is called the Boolean Gaussian component for µ, and the measure ν is called the Boolean Lévy measure for µ.The triplet (a, ν, γ) is called the Boolean Lévy triplet.
Let µ ∈ P (R) and ν be its Boolean Lévy measure.Let ν ac be the Lebesgue absolutely continuous part of ν.We introduce functions k µ , ℓ µ : R \ {0} → [0, ∞) by Those functions are key ingredients in our main results.Then ν ac (dx) can be expressed in the form

Free convolution and analytic tools
The free convolution of general probability measures on R was defined in [7].According to [7, Proposition 5.4], for any µ ∈ P (R) and any λ > 0 there exist α, β, M > 0 such that F µ is univalent on the set ).This implies that the right inverse function F −1 µ of F µ exists on Γ λ,M .The Voiculescu transform ϕ µ is defined by For µ, ν ∈ P (R), their free convolution µ ⊞ ν is the probability measure satisfying for z in the common domain in which the three transforms are defined.
The R-transform of µ ∈ P (R) is defined by By definition, it is obvious to see that R µ⊞ν (z) = R µ (z) + R ν (z) for all z in the common domain in which the three transforms are defined.
A probability measure µ on R is said to be freely infinitely divisible (denoted by µ ∈ I(⊞)) if for each n ∈ N there exists . Several criteria for free infinite divisibility were given by using harmonic analysis, complex analysis, and combinatorics (see e.g.[3,1]).In particular, the following characterization of I(⊞) is well-known (see [7,Theorem 5.10]).Proposition 2.2.For µ ∈ P (R), the following conditions are equivalent.
(2) The Voiculescu transform ϕ µ has an analytic extension (denoted by the same symbol ϕ µ ) defined on C + with values in C − ∪ R.
(3) There exist b ∈ R and a finite positive measure τ on R such that Note that such a pair (b, τ) is uniquely determined by µ for the same reason as (2.2)-(2.3).Conversely, given b ∈ R and a finite positive measure τ on R, there exists µ ∈ I(⊞) such that (2.9) holds.
The above pair (b, τ) is called the free generating pair for µ, and we denote by µ (b,τ) ⊞ the freely infinitely divisible distribution with a free generating pair (b, τ).
For µ ∈ I(⊞) with a free generating pair (b, τ), formula (2.9) is equivalent to where the triplet (a, ν, γ) is given by (2.5), see [2] for further details.The real number a ≥ 0 is called the free Gaussian component for µ, and the measure ν is called the free Lévy measure for µ.

Boolean-to-free Bercovici-Pata bijection
In [5, Theorem 6.3], Bercovici and Pata found a remarkable equivalence between limit theorems for convolutions * , ⊞ and ⊎.It leads to a bijection between the corresponding three classes of infinitely divisible distributions.Bercovici and Pata's work was concerning limit theorems for i.i.d.random variables.We present here a statement in a generalized setting of infinitesimal triangular arrays, combining [6, Theorem 1], [9, Theorem 2.1] and [22,Theorem 5.3].We use the notation µ n w − → µ to mean that µ n converges weakly to µ when µ n , µ are finite positive measures on R.
Theorem 2.3.Let {a n } n≥1 be a sequence of real numbers and {µ n,k } 1≤k≤k n ,1≤n be a family (or an array) of probability measures on R such that {k n } n≥1 is a sequence of natural numbers which tends to infinity and We set for Borel subsets B of R. Then the following conditions are equivalent. ( (4) there exist b ∈ R and a positive finite measure τ on R such that

Moreover, if one of those statements holds then
, where the last measure is the infinitely divisible distribution characterized by the Lévy-Khintchine representation From this result, it is natural to identify the limit distributions µ, µ ′ , µ ′′ .For later use, we only formulate the map Λ B between the two classes P (R) and I(⊞) sending µ ( ( (5) Λ B is a homeomorphism with respect to weak convergence.
Theorem 2.3 implies the following equivalence of Lévy's limit theorems (the classical one is already mentioned in Introduction).Although this will not be directly used in our paper, this will serve as a good motivation for studying Boolean selfdecomposable distributions.
Then the following conditions are equivalent. (1) The possible limit distributions µ, µ

Boolean selfdecomposable distributions 3.1 The class of Boolean selfdecomposable distributions
The classical notion of selfdecomposability can be extended to general convolutions of probability measures in the following way.
Definition 3.1.Let • be a binary operation on P (R).A measure µ ∈ P (R) is said to be •-selfdecomposable It is known that the free selfdecomposability has the following characterization (see e.g.[14, Subsection 2.2]).

Lemma 3.2. A probability measure µ on R belongs to the class L(⊞) if and only if µ is a freely infinitely divisible distribution of which free Lévy measure ν is Lebesgue absolutely continuous and the function
has a version with respect to the Lebesgue measure that is unimodal with mode 0, i.e. non-decreasing on (−∞, 0) and non-increasing on (0, ∞).
By the definition of Λ B , we obtain a characterization for the Boolean selfdecomposability exactly in the same way as Lemma 3.2, in which the free Lévy measure is to be replaced with the Boolean Lévy measure.

Proposition 3.3. A probability measure µ on R belongs to class L(⊎) if and only if its Boolean Lévy measure is
Lebesgue absolutely continuous and the function k µ in (2.7) has a version with respect to the Lebesgue measure that is unimodal with mode 0.
When we consider µ ∈ L(⊎), for simplicity we always take k µ itself to be unimodal with mode 0 unless specified otherwise.
Given a probability measure µ, a typical method for proving it to be Boolean selfdecomposable is Proposition 3.3.To check that the function k µ is unimodal with mode 0, we can calculate k µ from Proposition 3.4.To check that the Boolean Lévy measure is Lebesgue absolutely continuous, we provide a practical sufficient condition in Proposition 3.5.Proposition 3.4.Suppose that µ ∈ P (R).The function k µ : R \ {0} → [0, ∞) defined by (2.7) is then given by and the Boolean Gaussian component of µ is given by − lim ε→0 + iεF µ (iε).
Proof.Both formulas are basic.Let ν be the Boolean Lévy measure for µ and (b, τ) be the Boolean generating pair for µ.By Proposition 2.1 and relation (2.5), the F-transform of µ has the form  is contained in C and hence is at most countable.This implies that ρ sing is a discrete measure.Then assumption (3) implies that ρ does not have an atom in C, so that ρ sing = 0. Remark 3.6.It is clear that L(⊎) L(⊞) since the Bernoulli distribution 1  2 (δ The following example shows that L(⊞) L(⊎).Let f 1/2 be the positive free 1/2-stable distribution defined by where the square root above is defined as the principal branch (see [5,Appendix] for details).Stability implies selfdecomposability, so that f 1/2 ∈ L(⊞).To see By Proposition 3.5, the Boolean Lévy measure is Lebesgue absolutely continuous.By Proposition 3.4 the function k f 1/2 can be calculated into Because k f 1/2 is not non-increasing on (0, ∞), the desired conclusion f 1/2 / ∈ L(⊎) follows.

Example 3.8. The probability measure b α,ρ characterized by
is called a Boolean (strictly) stable distribution, where the parameter (α, ρ) belongs to the set

Regularity for Boolean selfdecomposable distributions
A general regularity result on boolean selfdecomposable distributions is established as follows.Note that we exclude the well understood Boolean Gaussian distributions (see Example 3.7) from the statement in order to have a non-empty support for the function k µ .
In particular, µ has no singular continuous part with respect to the Lebesgue measure and has at most two atoms.
Remark 3.10.The delta measures and Boolean Gaussian distributions are obviously singular distributions.Oher Boolean selfdecomposable distributions may also have a non-zero singular part, see e.g.
Remark 3.11.In the setting of Theorem 3.9, the points α and β may or may not be an atom of µ.To see this let µ be the Boolean selfdecomposable probability measure defined by where p > 0 is a parameter and b ∈ R is defined so that F µ (1 Example 3.12 (Mixture of Cauchy distribution and δ 0 ).

Its reciprocal Cauchy transform is given by
By Proposition 3.5, the Boolean Lévy measure is Lebesgue absolutely continuous.By Proposition 3.4, the function As long as p ∈ (0, 1), this is not unimodal with mode 0 and hence κ p is not Boolean selfdecomposable (cf.Theorem 3.9 (3)).
According to Theorem 3.9, every Boolean selfdecomposable distribution µ with k µ = 0 has at most two atoms.Here we completely determine the two point measures which are Boolean selfdecomposable.

Proposition 3.13. A two point probability measure on R belongs to L(⊎) if and only if it is a Boolean Gaussian distribution.
Proof.The Boolean Gaussian distribution B(γ, a) is a Boolean selfdecomposable two point probability measure for all γ ∈ R and a > 0, see (3.4).
Conversely, we assume that a two point probability measure µ belongs to L(⊎).By Proposition 3.4 we get k µ (x) = 0 for a.e.x ∈ R \ {0} because F µ is of the form (z 2 + pz + q)/(z − r), p, q, r ∈ R. Therefore µ has a Boolean Lévy triplet (a, 0, γ) for some γ ∈ R and some a ≥ 0. The Boolean Gaussian component a is nonzero; otherwise µ would be a Dirac measure.Consequently, we have µ = B(γ, a) for γ ∈ R and a > 0.

Boolean selfdecomposability of shifted probability measures
For any a ∈ R, "the Boolean shift" µ → µ ⊎ δ a preserves the class L(⊎).On the other hand, the usual shift µ → µ * δ a does not preserve L(⊎), which can be observed from Proposition 3.13.This phenomenon is investigated in details below.The function ℓ λ defined in (2.7) plays a key role.
Lemma 3.14.Suppose that λ ∈ P (R) satisfies the condition (C) the Boolean Gaussian component is zero and the Boolean Lévy measure is Lebesgue absolutely continuous.
Then for any m ∈ R the measure λ * δ m also satisfies condition (C) and Proof.According to (2.7), formula (3.3) can be expressed in the form where a is the Boolean Gaussian component, which is now 0. This yields for some b m ∈ R. We can therefore conclude that the Boolean Gaussian component of λ * δ m is zero and its Boolean Lévy measure equals t −2 ℓ λ (t − m)1 R\{0} (t) dt, as desired.
For a clear statement, we define Q to be the set of probability measures λ on R such that (H1) the Boolean Gaussian component of λ is zero, (H2) the Boolean Lévy measure of λ admits the form t −2 ℓ λ (t)1 R\{0} (t) dt, where ℓ λ : R → [0, ∞) is right continuous on R \ {m λ } for some m λ ∈ R.  ( Proof. (1) Suppose first that λ does not satisfy (H1), i.e. it has a positive Boolean Gaussian component a.
Similarly to (3.8), we obtain for some b m ∈ R, where ν is the Boolean Lévy measure of λ.This implies that the Boolean Lévy measure of λ * δ m is given by am −2 δ m + t −2 (t − m) 2 dν(t − m) as soon as m = 0.Because this is not Lebesgue absolutely continuous, λ * δ m is not Boolean selfdecomposable.
To complete the proof of (1), it suffices to prove that if λ satisfies (H1) and λ * δ m 0 ∈ L(⊎) for some m 0 ∈ R then λ satisfies (H2).According to Lemma 3.14, λ has a Lebesgue absolutely continuous Boolean Lévy measure and Because k λ * δ m 0 is unimodal with mode 0, it has a version that is right continuous on R \ {0}.Therefore, ℓ λ has a version that is right continuous on R \ {−m 0 }, i.e. (H2) holds.
(4) The distribution λ * δ m is then a delta measure (when ℓ λ = 0) or Cauchy distribution (when ℓ λ is a positive constant function) for any m ∈ R, which is Boolean selfdecomposable.Remark 3.16.In case (2) it might happen that λ * δ m is in L(⊎) for all sufficiently small m.To detect such an example, the function ℓ λ needs to be non-decreasing on R; otherwise case (3) would apply.Let λ be a probability measure with Boolean Lévy triplet (0, k λ (t) dt, 0), where Using the above lemma, we can explicitly compute the functions G N(0,1) and k N(0,1) .
Analyzing the function k N(0,1) we are able to demonstrate the following.
For the upper bound of (4.4), we use the following supplementary inequality This can be easily verified with calculus: the function H(x) := (e x 2 − 1)/x − h(x) satisfies H ′ > 0 on (0, ∞) and H(+0) = 0. Thanks to (4.5), for the upper bound of (4.4) it suffices to show that The latter is obvious if e x 2 − 2 < 0. In the case e x 2 − 2 ≥ 0 the desired inequality is equivalent to By calculus, this is the case.Thus we are done.
The lower bound of (4.4) can also be proved by calculus.Let It is elementary to see that J(+0) = 0.It then suffices to show that J ′ > 0. To begin, we compute Hence J ′ is obviously positive on [1/ √ 2, ∞).Suppose that J ′ (x) = 0 held for some 0 < x < 1/ √ 2. This would imply x 2 e −2x However, by some elementary calculus the left hand side of the last equation must be negative, a contradiction.We are done.
Next, we find a failure of Boolean selfdecomposablity for normal distributions.According to simulations, it is likely that ℓ ′′ N(0,1) in (−∞, 0) has a unique zero (denoted a 0 below) and hence p takes its minimum at a 0 ; see Figure 1.The values a 0 and M 0 = p(a 0 ) are approximately −2.03 and 3.09, respectively.Moreover, simulations also suggest that M 0 is the precise threshold, i.e.N(m, 1) is Boolean selfdecomposable if and only if |m| ≤ M 0 ; see Figures 2, 3.Those simulations are performed on Mathematica Version 12.1.1,Wolfram Research, Inc., Champaign, IL.

Lemma 2 . 4 .
(b,τ) ⊎ ∈ P (R) to µ (b,τ) ⊞ ∈ I(⊞) for each b ∈ R and each finite positive measure τ on R.This map Λ B is obviously a bijection and is called the Boolean-to-free Bercovici-Pata bijection.It turns out that Λ B has the following properties.Let µ 1 , µ 2 , µ ∈ P (R) and c ∈ R.

Figure 2 :Figure 3 :
Figure 1: the function p Corollary 2.5.Let {a n } n≥1 be a sequence of real numbers, {b n } n≥1 a sequence of positive real numbers and {µ n } n≥1 a sequence of probability measures on R such that (2.11) is fulfilled for the array µ n,k ′ , µ ′′ in Corollary 2.5 belong to certain subclasses of freely, Boolean and classically infinitely divisible distributions, respectively.Those subclasses are all characterized by the respective notions of selfdecomposability defined later in Definition 3.1.This fact follows e.g. from the last part of Theorem 2.3, [11, Theorem 1, Section 29] and [2, Theorem 4.8] (or [9, Theorem 2.10]) for * and ⊞.The case ⊎ can be treated similarly to (actually easier than) the free case.See also the paragraph following Definition 3.1.