STUDIA MATHEMATICA
Online First version

On Boolean selfdecomposable distributions
by

Takahiro Hasebe (Sapporo), Kei Noba (Tokyo),
Noriyoshi Sakuma (Nagoya) and Yuki Ueda (Asahikawa)
Abstract. 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 2 and the singular
continuous part is 0. 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|.

1. 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 randomvariables. From a certain viewpoint, notions of independence are classified
into five kinds: tensor, free, Boolean, monotone and anti-monotone independence (see [M03]). Further, each notion of independence has the associatedconvolution of (Borel) probability measures on R that is defined to be the
distribution of the sum of two independent (selfadjoint) random variables
having prescribed distributions.
Limit theorems have been central subjects in both commutative (classical) and non-commutative probability theories. For instance, Khinchin 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 asequence {Xn }n≥1 of independent R-valued random variables and sequences{an }n≥0 ⊂ R and {bn }n≥1 ⊂ (0, ∞) of deterministic numbers such that• the family {bn Xk }n≥k≥1 forms an infinitesimal triangular array, i.e.
lim

sup P [|bn Xk | ≥ ε] = 0

for all ε > 0,n→∞ 1≤k≤n2020 Mathematics Subject Classification: Primary 46L54; Secondary 60E07.
Key words and phrases: Boolean selfdecomposable distributions, Boolean convolution.
Received 27 January 2022; revised 5 October 2023.
Published online 24 January 2024.
DOI: 10.4064/sm221227-29-10[1]© Instytut Matematyczny PAN, 2024

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T. Hasebe et al.

• the law of the sequence of random variables
Sn = an + bn (X1 + · · · + Xn ),n ≥ 1,

converges weakly to µ.
When {Xn }n≥1 is furthermore identically distributed, the limit distribution µ, if it 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 L by the following property (see e.g. [GK54, Section 29, Theorem 1]):
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 µc on R such thatµ = Dc (µ)∗µc , where Dc (µ) is the push-forward of µ by the mapping x 7→ cx,
and ∗ denotes the classical convolution (see e.g. [Sat13]).
In non-commutative probability theory, an analogous limit theorem canbe formulated for each notion of independence. Bercovici and Pata [BP00],
Chistyakov and Götze [CG08] and Wang [W08] proved a parallelism between
classical Khinchin limit theorem above and its non-commutative versions (forfurther 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 detail (see e.g.
[BNT02, HST19, HT16] and [FHS20], respectively, where analogy and disanalogy with the classical class L are discussed). By contrast, little has been
done on Boolean selfdecomposable distributions. Although our definition of
the latter is natural for specialists, there has been no formal definition in theliterature 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 thatare used in the remainder of the paper. In Section 3, we establish a generalregularity result on Boolean selfdecomposable distributions. Specifically, wedemonstrate that they 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 it is typically lost when the distribution isshifted by 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 that section are motivated by the fact that every normal
distribution is both classically and freely selfdecomposable (see e.g. [Sat13]
and [HST19], 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

Boolean selfdecomposable distributions

3

consequence of the aforementioned general result in Section 3. Simulations
suggest that |m| ≃ 3.09 is the approximate threshold.
2. Preliminaries. In the first part of this section, we introduce analytic
tools for understanding Boolean and free additive convolutions. Then wesummarize a transfer principle for limit theorems for convolutions, which
induces a bijection between different kinds of infinitely divisible distributions.
2.1. Boolean convolution and analytic tools. The Boolean convolution of probability measures on R was defined in [SW97]. 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 half-plane. For µ ∈ P(R), we define
the Cauchy transform Gµ : C+ → C− and the F -transform Fµ : C+ → C+
as follows: 1
1
Gµ (z) :=µ(dt) and Fµ (z) :=, z ∈ C+ .
z−t
Gµ (z)
R

The self-energy function Kµ of µ is defined by
Kµ (z) := z − Fµ (z),z ∈ C+ .

Since Im[Fµ (z)] ≥ Im[z] for z ∈ C+ (see [BV93, Corollary 5.3]), the function
Kµ is an analytic map from C+ to C− ∪ R. The self-energy functions arecharacterized in the following way (see [SW97, 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 = Kµ .
(2) There exist b ∈ R and a finite positive measure τ on R, such that K has
the form 1 + xz
(2.1)
τ (dt),z ∈ C+ .
K(z) = b +z−xRIf 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 Proposition 2.1, there is a one-to-one correspondence between P(R) and the set(b,τ )of all Boolean generating pairs. Hence, we denote by µ⊎ the probability
measure µ with Boolean generating pair (b, τ ). The Boolean generating pair4

T. Hasebe et al.
(b,τ )

for µ⊎

can be computed from the following formulas:

(2.2)
(2.3)

Kµ(b,τ ) (i) = b − iτ (R),
⊎

β


β

1
(1 + t ) τ (dt) = − lim Im[Kµ(b,τ ) (x + iε)] dx⊎
π ε→0+ αα2for all α, β ∈ R that are continuity points of τ . The latter formula is referred
to as the Stieltjes inversion formula, due toεIm[Kµ(b,τ ) (x + iε)] = −· (1 + t2 ) τ (dt).
⊎(x − t)2 + ε2R

For µ, ν ∈ P(R), their Boolean convolution µ ⊎ ν is the probability measure on R satisfying(2.4)

Kµ⊎ν (z) = Kµ (z) + Kν (z),z ∈ C+ .

The η-transform of µ ∈ P(R) is defined by 
1, z ∈ C− .
ηµ (z) := 1 − zFµz
It is obvious that ηµ (z) = zKµ (1/z), and therefore ηµ⊎ν (z) = ηµ (z) + ην (z)for all z ∈ C− .
A probability measure µ is said to be Boolean infinitely divisible if foreach n ∈ N there exists µn ∈ P(R) such that µ = µn ⊎ · · · ⊎ µn (n times).
For each µ ∈ P(R) with Boolean generating pair (b, τ ), set µn to be theprobability measure with Boolean generating pair (b/n, τ /n) for each n ∈ N.
Then µ is the n-fold Boolean convolution of µn . Therefore every probability
measure on R is Boolean infinitely divisible.
For a probability measure µ on R with Boolean generating pair (b, τ ), let
us seta = τ ({0}),






1 + t2ν(dt)=· 1R\{0} (t)τ (dt),
(2.5)
t2





1



γ = b + t 1[−1,1] (t) − 1 + t2 ν(dt).
R

The triplet (a, ν, γ) thus defined satisfies(T) a ≥ 0, γ ∈ R and ν is a positive measure on R such that ν({0}) = 0
and R (1 ∧ x2 ) ν(dx) < ∞.
Moreover, the set of triplets (a, ν, γ) satisfying (T) is in bijection with theset of pairs (b, τ ) of a real number b and a finite positive measure τ on R. In

Boolean selfdecomposable distributions

5

terms of this bijection, formula (2.1) has the following equivalent form:

 
 1
1
2
(2.6) ηµ (z) = zKµ= γz + az +− 1 − zt1[−1,1] (t) ν(dt).
z
1 − ztR

The real number a ≥ 0 is called the Boolean Gaussian component for µ, andthe 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 kµ , ℓµ : R \ {0} →[0, ∞) bydν ackµ (x) = |x|(2.7)(x) and ℓµ (x) = |x|kµ (x), x ∈ R \ {0}.
dxThose functions are key ingredients in our main results. We observe thatν ac (dx) can be expressed in the form x−2 ℓµ (x) dx = |x|−1 kµ (x) dx.
Remark 2.2. The pair (kµ (x),ℓµ (x)) is determined uniquely in the senseof Remark 15.12(ii) in the book by Sato [Sat13].
2.2. Free convolution and analytic tools. The free convolution ofgeneral probability measures on R was defined in [BV93]. According to[BV93, Proposition 5.4], for any µ ∈ P(R) and any λ > 0 there exist
α, β, M > 0 such that Fµ is univalent on the set Γα,β := {z ∈ C+ : Im[z] > β,|Re[z]| < α Im[z]} and Γλ,M ⊂ Fµ (Γα,β ). This implies that the right inverse
function Fµ−1 of Fµ exists on Γλ,M . The Voiculescu transform φµ is definedbyφµ (z) := Fµ−1 (z) − z,z ∈ Γλ,M .

For µ, ν ∈ P(R), their free convolution µ ⊞ ν is the probability measure satisfying(2.8)

φµ ⊞ ν (z) = φµ (z) + φν (z)for z in the common domain in which the three transforms are defined.
The R-transform of µ ∈ P(R) is defined by 
11Rµ (z) := zφµ, z ∈ C− with ∈ Γλ,M .
zz
It is obvious that Rµ ⊞ ν (z) = Rµ (z) + Rν (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 µn ∈ P(R) such thatµ = µn ⊞ · · · ⊞ µn (n times). Several criteria for free infinite divisibility were
given by using harmonic analysis, complex analysis, and combinatorics (seee.g. [BBLS11, AH13]). In particular, the following characterization of I(⊞)
is well known (see [BV93, Theorem 5.10]).

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Proposition 2.3. For µ ∈ P(R), the following conditions are equivalent.
(1) µ ∈ I(⊞).
(2) The Voiculescu transform φµ has an analytic extension (denoted by thesame symbol φµ ) defined on C+ with values in C− ∪ R.
(3) There exist b ∈ R and a finite positive measure τ on R such that
 1 + tz
φµ (z) = b +τ (dt),z ∈ C+ .
(2.9)
z−t
R

Note that such a pair (b, τ ) is uniquely determined by µ for the same reasonas (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(b,τ )by µ⊞ the freely infinitely divisible distribution with free generating pair(b, τ ).
For µ ∈ I(⊞) with free generating pair (b, τ ), formula (2.9) is equivalent to 1(2.10) Rµ (z) = γz + az 2 +− 1 − zt1[−1,1] (t) ν(dt), z ∈ C− ,1 − ztR

where the triplet (a, ν, γ) is given by (2.5); see [BNT02] for further details.
The real number a ≥ 0 is called the free Gaussian component for µ, and themeasure ν is called the free Lévy measure for µ.
2.3. Boolean-to-free Bercovici–Pata bijection. In [BP99, Theorem
6.3], Bercovici and Pata found a remarkable equivalence between limit theorems for the convolutions ∗, ⊞ and ⊎. It leads to a bijection between thecorresponding three classes of infinitely divisible distributions. Bercovici andPata’s work concerned limit theorems for i.i.d. random variables. We present
here a statement in a generalized setting of infinitesimal triangular arrays,
combining [BP00, Theorem 1], [CG08, Theorem 2.1] and [W08, Theorem
5.3]. When µn , µ are finite positive measures on R, we use the notationwµn −→ µ to mean that µn converges weakly to µ.
Theorem 2.4. Let {an }n≥1 be a sequence of real numbers and let{µn,k }1≤k≤kn , n≥1 be a family (or an array) of probability measures on R
such that {kn }n≥1 is a sequence of natural numbers which tends to infinity
and(2.11)

lim

sup µn,k (R \ (−ε, ε)) = 0

for all ε > 0.

n→∞ 1≤k≤kn

We setan,k =

t µn,k (dt)
|x|<1

and

µ̊n,k (B) = µn,k (B + an,k )Boolean selfdecomposable distributions

7

for Borel subsets B of R. Then the following conditions are equivalent:w→ µ for some µ ∈ P(R) as n → ∞.
(1) δan ⊎ µn,1 ⊎ µn,2 ⊎ · · · ⊎ µn,kn −w→ µ′ for some µ′ ∈ P(R) as n → ∞.
(2) δan ⊞ µn,1 ⊞ µn,2 ⊞ · · · ⊞ µn,kn −w(3) δan ∗ µn,1 ∗ µn,2 ∗ · · · ∗ µn,kn −→ µ′′ for some µ′′ ∈ P(R) as n → ∞.
(4) There exist b ∈ R and a positive finite measure τ on R such that


kn 
 t
X
lim an +an,k + 2µ̊n,k (dt)
= b,
n→∞
t +1
k=1

kn
X
k=1

t2
t2 + 1Rwµ̊n,k (dt) −→ τ (dt)

as n → ∞.
(b,τ )(b,τ )

Moreover, if one of those statements holds then µ = µ⊎ , µ′ = µ⊞and(b,τ )′′µ = µ∗ , where the last measure is the infinitely divisible distribution
characterized by the Lévy–Khinchin representation





zt
1 + t2zt (b,τ )zte µ∗ (dt) = exp bz + e − 1 −τ (dt) ,z ∈ iR.
1 + t2t2
R

R

From this result, it is natural to identify the limit distributions µ, µ′ , µ′′ .
For later use, we only formulate the map ΛB between the two classes P(R)(b,τ )(b,τ )and I(⊞) sending µ⊎∈ P(R) to µ⊞∈ I(⊞) for each b ∈ R and eachfinite positive measure τ on R. This map ΛB is obviously a bijection and iscalled the Boolean-to-free Bercovici-Pata bijection. It turns out that ΛB has
the following properties.
Lemma 2.5. Let µ1 , µ2 , µ ∈ P(R) and c ∈ R. Then:(1) RΛB (µ) = ηµ on C− .
(2) ΛB (µ1 ⊎ µ2 ) = ΛB (µ1 ) ⊞ ΛB (µ2 ).
(3) ΛB (Dc (µ)) = Dc (ΛB (µ)).
(4) ΛB (δc ) = δc .
(5) ΛB is a homeomorphism with respect to weak convergence.
Formulas (1)–(4) can be checked by straightforward algebraic calculations, while assertion (5) needs careful analysis. The reader is referred to[BN08, (2.20)] for formula (1) and to [BNT02] for the other assertions.
Theorem 2.4 implies the following equivalence of Lévy’s limit theorems(the classical one is already mentioned in the Introduction). Although thiswill not be directly used in our paper, it will serve as a good motivation forstudying Boolean selfdecomposable distributions.
Corollary 2.6. Let {an }n≥1 be a sequence of real numbers, {bn }n≥1 be
a sequence of positive real numbers and {µn }n≥1 a sequence of probability

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T. Hasebe et al.

measures on R such that (2.11) is satisfied for the array µn,k := Dbn (µk ),1 ≤ k ≤ n, 1 ≤ n. Then the following conditions are equivalent:w(1) δan ⊎ Dbn (µ1 ⊎ · · · ⊎ µn ) −→ µ for some µ ∈ P(R) as n → ∞.
w
→ µ′ for some µ′ ∈ P(R) as n → ∞.
(2) δan ⊞ Dbn (µ1 ⊞ · · · ⊞ µn ) −w→ µ′′ for some µ′′ ∈ P(R) as n → ∞.
(3) δan ∗ Dbn (µ1 ∗ · · · ∗ µn ) −The possible limit distributions µ, µ′ , µ′′ in Corollary 2.6 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 2.4, [GK54, Section 29, Theorem 1] and[BNT02, Theorem 4.8] (or [CG08, Theorem 2.10]) for ∗ and ⊞. The case of ⊎can be treated similarly to (and is actually easier than) the free case. See
also the paragraph following Definition 3.1.
3. 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). Then a measureµ ∈ P(R) is said to be ◦-selfdecomposable (denoted µ ∈ L(◦)) if for anyc ∈ (0, 1) there is µc ∈ P(R) such that µ = Dc (µ) ◦ µc . In particular, if◦ = ⊎ (resp. ◦ = ⊞), then µ is said to be Boolean selfdecomposable (resp.
freely selfdecomposable).
By Lemma 2.5(2)–(3) and [BNT02, Theorem 4.6, Proposition 4.7], we
have
(3.1)

ΛB (L(⊎)) = L(⊞) and L(⊎) = Λ−1
B (L(⊞)).

It is known that the free selfdecomposability has the following characterization (see e.g. [HT16, Subsection 2.2]).
Lemma 3.2. A probability measure µ on R belongs to the class L(⊞) if andonly if µ is a freely infinitely divisible distribution whose free Lévy measureν is Lebesgue absolutely continuous and the function R \ {0} → [0, ∞), x 7→dν
|x| dx(x), 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 ΛB , we obtain a characterization for the Boolean
selfdecomposability exactly in the same way as Lemma 3.2, in which the free
Lévy measure is replaced with the Boolean Lévy measure.
Proposition 3.3. A probability measure µ on R belongs to the class L(⊎)
if and only if its Boolean Lévy measure is Lebesgue absolutely continuous andBoolean selfdecomposable distributions

9

the function kµ in (2.7) has a version with respect to the Lebesgue measurethat is unimodal with mode 0.
When we consider µ ∈ L(⊎), for simplicity we always take kµ 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 kµ
is unimodal with mode 0, we can calculate kµ 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 kµ : R \ {0} →[0, ∞) defined by (2.7) is then given by1kµ (x) = lim(3.2)
Im[Fµ (x + iε)] for a.e. x ∈ R \ {0}+ε→0 π|x|and the Boolean Gaussian component of µ is given by − limε→0+ iεFµ (iε).
Proof. Both formulas are basic. Let ν be the Boolean Lévy measure for µand (b, τ ) be the Boolean generating pair for µ. By Proposition 2.1 andrelation (2.5), the F-transform of µ has the form
  1 + t2
(3.3)
Fµ (z) = z − b −+ t τ (dt)
z−t
R


1
a 
t
=z−b− +−
|t|2 ν(dt).
zt − z 1 + t2
R

Since the Lebesgue absolutely continuous part of |t|2 ν(dt) is |t|kµ (t) dt, thedesired formula for kµ is a consequence of the Stieltjes inversion formula(see e.g. [HO07, Corollary 1.103] or [Sch12, Theorem F.6]). The formula for
the Boolean Gaussian component follows from the Dominated Convergence
Theorem.
Proposition 3.5. Let µ ∈ P(R). Suppose that there exists an at mostcountable subset C of R such that(1) 0 ∈ C,(2) limε→0+ Im[Fµ (x + iε)] ∈ [0, ∞) exists for all x ∈ R \ C,(3) limε→0+ εFµ (x + iε) = 0 for all x ∈ C \ {0}.
Then the Boolean Lévy measure of µ is Lebesgue absolutely continuous.
Proof. Let ν be the Boolean Lévy measure of µ and ρ(dt) := |t|2 ν(dt).
Note that ρ({0}) = 0. It suffices to show that ρ is Lebesgue absolutely continuous. Let ρ = ρsing + ρac be the decomposition of ρ into the singular partand the absolutely continuous part with respect to the Lebesgue measure.

10

T. Hasebe et al.

From formula (3.3) and [Sch12, Theorem F.6], ρsing is supported on the setnoSρsing := x ∈ R : lim Im[Fµ (x + iε)] = ∞ε→0+in the sense that ρsing (R \ Sρsing ) = 0. By assumption (2), the set Sρsing iscontained in C and hence is at most countable. This implies that ρsing is adiscrete measure. Then assumption (3) implies that ρ does not have an atom
in C, so that ρsing = 0.
Remark 3.6. It is clear that L(⊎) ⊈ L(⊞) since the Bernoulli distribution1
2 (δ1 + δ−1 ) is in L(⊎), but not in L(⊞).

The following example shows that L(⊞) ⊈ L(⊎). Let f1/2 be the positive
free 1/2-stable distribution defined by√
φf1/2 (z) = −z, z ∈ C+ ,where the square root above is defined as the principal branch (see [BP99,
Appendix] for details). Stability implies selfdecomposability, so f1/2 ∈ L(⊞).
√
(z)
=
z
+
To see f1/2 ∈/ L(⊎), inverting Ff−1
−z yields
1/2√2z − 1 − 1 − 4z
Ff1/2 (z) =, z ∈ C+ .
2
By Proposition 3.5, the Boolean Lévy measure is Lebesgue absolutely continuous. By Proposition 3.4 the function kf1/2 can be calculated as√
4x − 1· 1[1/4,∞) (x).
kf1/2 (x) =2πxSince kf1/2 is not non-increasing on (0, ∞), the desired conclusion f1/2 ∈/ L(⊎)
follows.
Example 3.7. The probability measure having the Boolean Lévy triplet(a, 0, γ) with a > 0 is called a Boolean Gaussian distribution and will bedenoted B(γ, a). It has mean γ, variance a and is of the form
(3.4)




1
1
γ
γ
.
− p
δ(γ−√γ 2 +4a2 )/2 ++ p
δ √2 2 γ 2 + 4a2
2 2 γ 2 + 4a2 (γ+ γ 2 +4a2 )/2
Because the Boolean Lévy measure is zero, B(γ, a) is Boolean selfdecomposable. Note that B(a, γ) can be written in the simpler form1(βδβ − αδα )β−αppwith parameters α = (γ − γ 2 + 4a2 )/2 < 0 and β = (γ + γ 2 + 4a2 )/2> 0. In fact, the map (γ, a) 7→ (α, β) gives a bijection from R × (0, ∞) to(−∞, 0) × (0, ∞).

Boolean selfdecomposable distributions

11

In particular, one can see that pδ1 + (1 − p)δ−1 for p ∈ (0, 1) is a Boolean
Gaussian distribution if and only if p = 1/2.
Example 3.8. The probability measure bα,ρ characterized byηbα,ρ (z) = −(eiρπ z)α ,z ∈ C− ,is called a Boolean (strictly) stable distribution, where the parameter (α, ρ)belongs to the set
A = {(α, ρ) : α ∈ (0, 1], ρ ∈ [0, 1]} ∪ {(α, ρ) : α ∈ (1, 2], ρ ∈ [1 − α−1 , α−1 ]}.
The Boolean stable distributions satisfy the relation bα,ρ = Dc (bα,ρ ) ⊎D(1−cα )1/α (bα,ρ ) for any c ∈ (0, 1), and hence are Boolean selfdecomposable.
Proposition 3.4 implies the formulasin(αρπ) −αsin(α(1 − ρ)π) −αkbα,ρ (x) =x 1(0,∞) (x) +|x| 1(−∞,0) (x).
ππThe special case b2,1/2 coincides with the Bernoulli distribution B(0, 1). For
further information on bα,ρ , see e.g. [SW97, HS15].
3.2. Regularity for Boolean selfdecomposable distributions. We
establish a general regularity result on Boolean selfdecomposable distributions. 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 kµ .
Theorem 3.9. Let µ ∈ L(⊎) with the function kµ not identically equal
to zero. Let α := inf {x ∈ R : kµ (x) ̸= 0} ∈ [−∞, 0] and β := sup {x ∈ R :kµ (x) ̸= 0} ∈ [0, ∞]. Then(1) µ|(α,β) is absolutely continuous with respect to the Lebesgue measure.
(2) µ|(−∞,α] is the zero measure or a delta measure if α > −∞. The same
holds for µ|[β,∞) if β < ∞.
(3) µ({0}) = 0.
In particular, µ has no singular continuous part with respect to the Lebesgue
measure and has at most two atoms.
Proof. Let us take the Pick–Nevanlinna representation of Fµ in the form

β
1
t
a 
−
|t|kµ (t) dt,(3.5)
Fµ (z) = z − b − +z α t − z 1 + t2
where a ≥ 0 and b ∈ R; see (3.3).
(2) It suffices to prove the statement for (−∞, α] by symmetry. We assume α > −∞. Formula (3.5) enables Fµ to be extended analytically toC+ ∪ (−∞, α). The extended function, still denoted Fµ , takes real values on(−∞, α); hence, the value of the function Im[Gµ (z)] = − Im[Fµ (z)]/|Fµ (z)|212

T. Hasebe et al.

is zero on (−∞, α) \ {zeros of Fµ |(−∞,α) }. By the Stieltjes inversion formula,
we conclude thatµ (−∞, α) \ {zeros of Fµ |(−∞,α) } = 0.
Because Fµ′ > 1 on (−∞, α), Fµ has at most one zero on (−∞, α).
If Fµ has no zeros on (−∞, α), then µ|(−∞,α] is the zero measure or adelta measure at α.
If Fµ has a zero x0 in (−∞, α), thenz − x01µ({x0 }) = lim= ′> 0.
z→x0 Fµ (z)Fµ (x0 )In this case we need to check that µ({α}) = 0. Because Fµ is strictly increasing and Fµ (x0 ) = 0, we have limx→α−0 Fµ (x) ∈ (0, ∞] and hence Gµ (α) :=limx→α−0 Gµ (x) ∈ [0, ∞). Directly or by Lindelöf’s theorem [BCDM20, Theorem 1.5.7], it can be verified that Gµ (α) = limε→0+ Gµ (α + iε). Therefore,µ({α}) = limε→0+ iεGµ (α + iε) = 0.
(3) Because kµ is unimodal and is not identically zero, we have either
α < 0 or β > 0. Without loss of generality we may work in the latter case.
Then δ := inf x∈(0,β/2) kµ (x) > 0. Because µ({0}) = limε→0+ iε/Fµ (iε), thedesired assertion µ({0}) = 0 follows from limε→0+ Im[Fµ (iε)]/ε = ∞. The
latter is the case becauseβ |t|k (t)Im[Fµ (iε)]
aµ=1+ 2 + 2
dt
ε
ε
t
+
ε2
α
≥δ

β/2

0

 1
t
dt
→
δ
dt = ∞
t2 + ε2tβ/2as ε → 0+ .

0(1) Let µ = µsing + µac be the decomposition of µ into the singularpart and absolutely continuous part with respect to the Lebesgue measure.
From (3), it suffices to prove µsing ((α, β) \ {0}) = 0. As in the proof of
Proposition 3.5, µsing (R \ S) = 0, wherenoS := x ∈ R : lim Im[−Gµ (x + iε)] = ∞ .
ε→0+Let us assume that β > 0; otherwise we need not work on the interval(0, β). Because kµ is unimodal with mode 0, for every 0 < β ′ < β the numberδ ′ := inf x∈(0,β ′ ] kµ (x) is positive. For every ε > 0 and x ∈ (0, β ′ ) we haveβa  ε|t|kµ (t)Im[Fµ (x + iε)] = ε + +dt
ε α (t − x)2 + ε2
′

≥δx

β′

ε
β′ − x′dt=
δxarctan,(t − x)2 + ε2εxBoolean selfdecomposable distributions

13

which tends to δ ′ xπ/2 as ε tends to 0. This implies that
Im[Fµ (x + iε)]|Fµ (x + iε)|2ε→0+
12≤ lim sup
≤ ′
<∞
Im[F
(x
+
iε)]
δ
xπ
+
µ
ε→0

lim sup Im[−Gµ (x + iε)] = lim sup
ε→0+for all x ∈ (0, β ′ ). We can thus conclude that (0, β) ⊆ R \ S, i.e. µsing ((0, β))= 0. Similarly or by symmetry, we get µsing ((α, 0)) = 0 if α < 0.
Remark 3.10. The delta measures and Boolean Gaussian distributions
are obviously singular distributions. Other Boolean selfdecomposable distributions may also have a non-zero singular part; see e.g. Example 3.17(3, 4).
By contrast, all classical selfdecomposable distributions and free distributions except for the delta measures are Lebesgue absolutely continuous (see[Sat13, Theorem 27.13] and [HT16, Remark 2], respectively).
Remark 3.11. In the setting of Theorem 3.9, the points α and β may
or may not be atoms of µ. To see this let µ be the Boolean selfdecomposable
probability measure defined by
Fµ (z) = z − b +101(1 − t)p
|t|
dt,
t−z
|t|

z ∈ C+ ,where p > 0 is a parameter and b ∈ R is defined so that Fµ (1) := 1 − b −1
p−1 dt = 0. One sees that
0 (1 − t) (1 − t)p−1
Fµ (z)Fµ (z) − Fµ (1)
=
=1+
dt
z−1
z−1
z−t
1
0

1

→ q := 1 + (1 − t)p−2 dt ∈ (1, ∞]0

as z → 1 satisfying Re[z] ≥ 1. This implies that µ({1}) = 1/q, which is
positive if p > 1 and zero if 0 < p ≤ 1.
Example 3.12 (Mixture of a Cauchy distribution and δ0 ). Let us setpκp =1R (x) dx + (1 − p)δ0π(1 + x2 )for p ∈ [0, 1]. Its reciprocal Cauchy transform is given byFκp (z) =1

z(z + i)p1−p = z + (1 − p)i ,z+i + zz ∈ C+ .

By Proposition 3.5, the Boolean Lévy measure is Lebesgue absolutely con-

14

T. Hasebe et al.

tinuous. By Proposition 3.4, the function kκp is given bykκp (x) =

1
p|x|
· 2
·1
(x).
π x + (1 − p)2 R\{0}As long as p ∈ (0, 1), this is not unimodal with mode 0 and hence κp is not
Boolean selfdecomposable (cf. Theorem 3.9(3)).
According to Theorem 3.9, every Boolean selfdecomposable distributionµ with kµ ̸= 0 has at most two atoms. Here we completely determine thetwo-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 belongsto L(⊎). By Proposition 3.4 we get kµ (x) = 0 for a.e. x ∈ R \ {0} because
Fµ 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 non-zero; otherwise µ would be a Dirac measure.
Consequently, we have µ = B(γ, a) for γ ∈ R and a > 0.
3.3. Boolean selfdecomposability of shifted probability measures. For any a ∈ R, “the Boolean shift” µ 7→ µ ⊎ δa preserves the classL(⊎). On the other hand, the usual shift µ 7→ µ ∗ δa does not preserve L(⊎),
which can be observed from Proposition 3.13. This phenomenon is investigated in detail below. The function ℓλ 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 ∗ δm also satisfies condition (C) and(3.6)ℓλ∗δm (t) = ℓλ (t − m)for a.e. t ∈ R.

Proof. According to (2.7), formula (3.3) can be expressed in the form

  1
a
t
Fλ (z) = z − b − +(3.7)−ℓλ (t) dt,z
t − z 1 + t2
R\{0}where a is the Boolean Gaussian component, which is now 0. This yields(3.8)

Fλ∗δm (z) = Fλ (z − m)
= z − bm +


  1
t
−
ℓλ (t − m) dt
t − z 1 + t2

R\{0}

Boolean selfdecomposable distributions

15

for some bm ∈ R. Therefore, the Boolean Gaussian component of λ ∗ δmis zero and its Boolean Lévy measure equals t−2 ℓλ (t − m)1R\{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 has the form t−2 ℓλ (t)1R\{0} (t) dt, whereℓλ : R → [0, ∞) is right continuous on R \ {mλ } for some mλ ∈ R.
Theorem 3.15.
(1) If λ ∈ P(R) \ Q then λ ∗ δm ∈/ L(⊎) for any m ∈ R \ {0}.
(2) If λ ∈ Q and there exist a, b ∈ R \ {mλ } with a < b and ℓλ (a) < ℓλ (b)λ (b)then λ ∗ δm ∈/ L(⊎) for any m > bℓℓλλ(a)−aℓ(b)−ℓλ (a) .
(3) If λ ∈ Q and there exist c, d ∈ R \ {mλ } with c < d and ℓλ (c) > ℓλ (d)λ (c)then λ ∗ δm ∈/ L(⊎) for any m < cℓℓλλ(d)−dℓ(c)−ℓλ (d) .
(4) If λ ∈ Q and ℓλ is constant on R \ {mλ } then λ ∗ δm ∈ L(⊎) for allm ∈ 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

  1
t
a
+
−
(t−m)2 dν(t−m)(3.9) Fλ∗δm (z) = z−bm +m−zt − z 1 + t2
R\{0}for some bm ∈ R, where ν is the Boolean Lévy measure of λ. This implies
that for m ̸= 0, the Boolean Lévy measure of λ ∗ δm is given by am−2 δm +t−2 (t − m)2 dν(t − m). Because this is not Lebesgue absolutely continuous,
λ ∗ δm is not Boolean selfdecomposable.
To complete the proof of (1), it suffices to prove that if λ satisfies (H1)and λ ∗ δm0 ∈ L(⊎) for some m0 ∈ R then λ satisfies (H2). According toLemma 3.14, λ has a Lebesgue absolutely continuous Boolean Lévy measureand(3.10)ℓλ (t) = ℓ(λ∗δm0 )∗δ−m0 (t) = ℓλ∗δm0 (t + m0 )= |t + m0 |kλ∗δm0 (t + m0 )for a.e. t ∈ R \ {0}.

Because kλ∗δm0 is unimodal with mode 0, it has a version that is right continuous on R \ {0}. Therefore, ℓλ has a version that is right continuous on
R \ {−m0 }, i.e. (H2) holds.
λ (b)(2) For all m > M := bℓℓλλ(a)−aℓ(b)−ℓλ (a) , observe that 0 < m + a < m + b and,by Lemma 3.14,ℓλ (a)ℓλ (b)(3.11)
kλ∗δm (m + a) − kλ∗δm (m + b) =−< 0.
m+a m+b

16

T. Hasebe et al.

Because ℓλ is right continuous at a and b, inequality (3.11) still holds whena, b are replaced with a + ε, b + ε for sufficiently small ε > 0; hence, for anym > M , kλ∗δm does not have a version that is non-increasing on (0, ∞), andconsequently λ ∗ δm is not in L(⊎). The proof of (3) is similar.
(4) In this case, the distribution λ ∗ δm is a delta measure (when ℓλ = 0)
or a Cauchy distribution (when ℓλ is a positive constant function) for anym ∈ R, which is Boolean selfdecomposable.
Remark 3.16. In case (2) it might happen that λ ∗ δm is in L(⊎) for all
sufficiently small m. To obtain such an example, the function ℓλ needs to be
non-decreasing on R; otherwise case (3) would apply. Let λ be a probability
measure with Boolean Lévy triplet (0, kλ (t) dt, 0), where1πkλ (t) =arctan t +, t ∈ R \ {0}.
|t|2
It can be checked by calculus that the function1πarctan(t − m) +, t ∈ R \ {0},kλ∗δm (t) :=|t|
2
is unimodal with mode 0 if and only if m ≤ π/2, and hence λ ∗ δm ∈ L(⊎)
for any such m.
3.4. Several examples of Boolean selfdecomposable distributions. We observe distributional properties of Boolean selfdecomposable
distributions through several examples.
Example 3.17. We give several examples in the class L(⊎).
(1) The free Poisson (or Marchenko–Pastur ) distribution MPλ is a probability measure defined by
1 p(λ+ − x)(x − λ− ) · 1(λ− ,λ+ ) (x) dx, λ > 0,
max{1 − λ, 0}δ0 +
2πx√where λ± = (1 ± λ)2 . Its F-transform is given bypz + 1 − λ + (z − λ− )(z − λ+ ), z ∈ C+ ,FMPλ (z) =2√where the square root w is defined continuously on the angles arg w ∈(0, 2π). By Propositions 3.5 and 3.4, the Boolean Lévy measure is Lebesgue
absolutely continuous and the function kMPλ is given byp(x − λ− )(λ+ − x)kMPλ (x) =· 1(λ− ,λ+ ) (x),2πxand therefore MPλ ∈ L(⊎) if and only if λ = 1. Consequently, L(⊎) is not
closed under the free convolution since MP1 ⊞ MP1 = MP2 ∈/ L(⊎), forexample.

Boolean selfdecomposable distributions

17(2) Let S(m, σ 2 ) be Wigner’s semicircle law, that is, its probability density function is defined by
1 p 2
4σ − (x − m)2 · 1[m−2σ,m+2σ] (x), m ∈ R, σ > 0.
2πσ 2Its F-transform is known to bepz − m + (z − m)2 − 4σ 2
, z ∈ C+ .
FS(m,σ2 ) (z) =2
By Proposition 3.5 the Boolean Lévy measure is Lebesgue absolutely continuous and by Proposition 3.4 we havep
4σ 2 − (x − m)2· 1[m−2σ,m+2σ]\{0} (x),kS(m,σ2 ) (x) =2π|x|and therefore S(m, σ 2 ) ∈ L(⊎) if and only if m − 2σ ≤ 0 ≤ m + 2σ.
(3) For any t > 1, the measure B(0, 1)⊞t is given byp
1 t 4(t − 1) − x22−t
, 0 (δt + δ−t ) +·· 1[−2√t−1,2√t−1] (x) dx.
max
22π
t 2 − x2
It is called the Kesten distribution. The F-transform is given byp(t − 2)z + t z 2 − 4(t − 1), z ∈ C+ .
FB(0,1)⊞t (z) =2(t − 1)
By Propositions 3.5 and 3.4, the Boolean Lévy measure is Lebesgue absolutely continuous andpt 4(t − 1) − x2
kB(0,1)⊞t (x) =· 1[−2√t−1,2√t−1]\{0} (x).
2π(t − 1)|x|
Because kB(0,1)⊞t is unimodal with mode 0 for all t > 1, we conclude that
B(0, 1)⊞t ∈ L(⊎) for all t > 1.
(4) Let µ(p, r) be the 2-parameter Fuss–Catalan distribution. Recall thatµ(p, r) ∈ I(⊞) if and only if either 0 < r ≤ min{p/2, p − 1} or 1 ≤ p = r ≤ 2(see [MSU20, Theorem 4.1]). Then we can define µe(p, r) := Λ−1
B (µ(p, r)) forsuch pairs (p, r). Since µ(p, r) ∈ L(⊞) if and only if 1 ≤ p = r ≤ 2 (see[MSU20, Theorem 4.3]), we have µe(p, p) ∈ L(⊎) for 1 ≤ p ≤ 2 by (3.1). In
particular, µe(2, 2) = B(2, 1) is a Boolean Gaussian distribution. By Lemma
2.5(1) and [MSU20, Proposition 3.6], for 1 ≤ p < 2 the Boolean Lévy tripletof µe(p, p) is given by (0, |x|−1 kµ(p,p) (x) dx, p), wheresin(pπ) 1 + x p
kµ(p,p) (x) = −· 1(−1,0) (x).
π−xFrom computations similar to [MSU20, Proposition 3.6], we obtainηµe(p,p) (z) = (1 + z)p − 1,z ∈ C− .

18

T. Hasebe et al.

Then the Cauchy transform is as follows:1
Gµe(p,p) (z) =, z ∈ C+ .
z[2 − (1 + z −1 )p ]By the Stieltjes inversion formula, the measure µe(p, p) for 1 ≤ p < 2 has thefollowing explicit form:psin(pπ) 1+x1xδ+· 1(−1,0) (x) dx,ap(2 − 21−1/p ) p πx(4 − 4 cos(pπ) 1+x p + 1+x 2p )

x
x
1/p
−1
(2
− 1) . According to this example, the discrete parts

where ap =of Boolean selfdecomposable distributions come from not only the Boolean
Gaussian component but also its Boolean Lévy measure.

4. Boolean selfdecomposability for normal distributions. In thissection, we investigate Boolean selfdecomposability for normal distributions
N (m, v). The question is unimodality of kN (m,v) with mode 0. It turns outthat kN (m,v) has an explicit formula. To compute it, we make use of thefollowing differential equation (see [K98, (8.1.6)]), which can be derived byintegration by parts.
Lemma 4.1. The following differential equation holds:
G′N (0,1) (z) = 1 − zGN (0,1) (z),z ∈ C+ .

Using the above lemma, we can explicitly compute the functions GN (0,1)
and kN (0,1) .
z

2Proposition 4.2. Let h(z) = 0 et dt, z ∈ C. Then
 r


z
π √
− 12 z 2−i+ 2h √,z ∈ C+ ,(4.1)
GN (0,1) (z) = e22
and the Boolean Lévy triplet for N (0, 1) is (0, kN (0,1) (x) dx, 0), wherer
1 2
1
kN (0,1) (x) =
·
(4.2)

  · 1R\{0} (x).
π π |x|e−x2 /2 1 + 4 h √x 2
π2

Proof. The standard variation of constants method for the differential
equation in Lemma 4.1 yields the solution
z
h
i
2
2
GN (0,1) (z) = e−z /2 C + et /2 dt , z ∈ C+ ,0

for some C ∈ C. Because N (0, 1) is symmetric about 0, Re[GN (0,1) (iy)] = 0
for all y > 0 and so Re[C] = 0. In order that the density of N (0, 1) is obtainedp
as the limit (−1/π) limε→0+ GN (0,1) (x + iε), we must have Im[C] = − π/2,
proving the desired formula (4.1). Because FN (0,1) has a continuous extension
to C+ ∪ R, the Stieltjes inversion (2.3) implies that the Boolean GaussianBoolean selfdecomposable distributions

19

component is zero and the Boolean Lévy measure is Lebesgue absolutely
continuous. By Proposition 3.4 we have (4.2). By (2.2) and Re[FN (0,1) (i)] = 0
we obtain b = 0. Finally, because τ (dx) =zero, (2.5) yields γ = 0.

|x|kN (0,1) (x)dx is symmetric about
1+x2

Analyzing the function kN (0,1) we are able to demonstrate the following.
Theorem 4.3. For all v > 0, we have N (0, v) ∈ L(⊎).
Proof. Since D√v (N (0, 1)) = N (0, v), it suffices to show N (0, 1) ∈ L(⊎).
To do so, we prove unimodality for kN (0,1) with mode 0 using Proposition 4.2.
Because the function kN (0,1) is symmetric with respect to 0, it is enough toprove that kN (0,1) (x) is non-increasing on (0, ∞). For this it suffices to showthat ℓN (0,1) is non-increasing on (0, ∞), i.e. the function


 
4
x 2
−x2 /2(4.3)
f (x) = e1+ h √π2
is non-decreasing on (0, ∞). For concise
calculations, let us instead work√with the rescaled function g(x) = f ( 2 x).
To begin we compute


8
4
2
′
−x2
g (x) = −2xe
1 + h(x) + h(x)
π
π


2
2
8xe−x
ex
π
=−
h(x)2 −
h(x) +.
πx
4The desired inequality, g ′ ≥ 0 on (0, ∞), is therefore equivalent to 2 s x2 2 2 s x2 2

1 ex
e
1 ex
e
(4.4)
−
− π ≤ h(x) ≤
+
−π .
2 x
x
2 x
x
2

Note here that (ex /x)2 ≥ 2e > π for all x > 0.
For the upper bound of (4.4), we use the supplementary inequality2ex − 1, x > 0.
(4.5)
h(x) <
x2
This can be easily verified with calculus: H(x) := (ex − 1)/x − h(x) satisfies
H ′ > 0 on (0, ∞) and H(+0) = 0. Thanks to (4.5), for the upper boundof (4.4) it suffices to show that x2 s x2 2

2
1 e
e
ex − 1
+
−π ≥
,
2 x
x
x
or equivalently
s

2
ex2 2
ex − 2
−π ≥
.
x
x

20

T. Hasebe et al.
22

The latter is obvious if ex − 2 < 0. In the case ex − 2 ≥ 0 the desired
inequality is equivalent to x2  222e2x − 4ex + 4
e
−π ≥
,
x
x2
or equivalently
24ex − πx2 − 4 ≥ 0.
By calculus, this is the case. Thus we are done.
The lower bound of (4.4) can also be proved by calculus. Let x2 s x2 2eeJ(x) := 2h(x) −
−
−π .
x
x
It is elementary to see that J(+0) = 0. It then suffices to show that J ′ > 0.
To begin, we compute
 x2

2
2
−2e2x /x3 + 4e2x /x
e
2
2
p 2
J ′ (x) = 2ex − − 2 + 2ex −x
2 (ex /x)2 − π2−1 + 2x2
e2x
−x2
+p 2
= 3 xe.
x(ex /x)2 − π√′Hence J ′ is obviously√ positive on [1/ 2, ∞). Suppose that J (x) = 0 held
for some 0 < x < 1/ 2. This would imply
 x2 2e
2 −2x2x e− π = (2x2 − 1)2 ,x
or equivalently2πe−2x + 4x2 − 4 = 0.
However, by elementary calculus the left hand side of the last equation must
be negative, a contradiction.
Next, we show the failure of Boolean selfdecomposablity for normal distributions.
√
Theorem 4.4. There exists M > 0 such that |m/ v| > M implies that
N (m, v) ∈/ L(⊎).
Proof. It suffices to find M > 0 such that N (m, 1) ∈/ L(⊎) for |m| > M
thanks to the fact that dilation preserves L(⊎). RecallcℓN (0,1) (x) = |x|kN (0,1) (x) =
,
f (x)
p
where c = π −1 2/π and f is defined in (4.3). Because N (0, 1) has no
Boolean Gaussian component by Proposition 4.2 and the function ℓN (0,1)

Boolean selfdecomposable distributions

21

is not constant on (0, ∞) and is symmetric on R, the desired conclusion
follows by Theorem 3.15.
Remark 4.5 (Estimate of M ). According to Theorem 3.15, N (m, 1) is
not in L(⊎) if |m| > M0 , wherebℓN (0,1) (a) − aℓN (0,1) (b) −∞ < a < b < ∞,(4.6)
M0 = inf.
:ℓN (0,1) (b) − ℓN (0,1) (a)ℓN (0,1) (a) < ℓN (0,1) (b)With some inspection helped by the identitybℓN (0,1) (a) − aℓN (0,1) (b)(b − a)ℓN (0,1) (a)=−aℓN (0,1) (b) − ℓN (0,1) (a)ℓN (0,1) (b) − ℓN (0,1) (a)=(b − a)ℓN (0,1) (b)− b,ℓN (0,1) (b) − ℓN (0,1) (a)the infimum (4.6) turns out to be achieved in the limit |b − a| → 0 and only
when a, b are both negative, so that we obtainℓN (0,1) (a)M0 = inf {p(a) : −∞ < a < 0}, where p(a) := ′− a.
ℓN (0,1) (a)In order to find the infimum of the function p, observe that−ℓN (0,1) (a)ℓ′′N (0,1) (a)′p (a) =.
ℓ′N (0,1) (a)2
According to simulations, it is likely that ℓ′′N (0,1) in (−∞, 0) has a uniquezero (denoted a0 below) and hence p takes its minimum at a0 ; see Figure 1.
14

12

10

8

6

4

2

-14

-12

-10

-8

-6

-4

-2

0

Fig. 1. p(a)The values a0 and M0 = p(a0 ) are approximately −2.03 and 3.09, respectively. Moreover, simulations also suggest that M0 is the precise threshold,i.e. N (m, 1) is Boolean selfdecomposable if and only if |m| ≤ M0 ; see Figures 2, 3. Those simulations are performed on Mathematica Version 12.1.1,
Wolfram Research, Inc., Champaign, IL.

22

T. Hasebe et al.
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4

0.2

0.2

-4

-2

2

4

-4

Fig. 2. kN (3.05,1) (a)

-2

2

4

Fig. 3. kN (3.2,1) (a)Acknowledgements. The authors express sincere thanks to the referee,
who gave useful comments to improve the manuscript.
This research is supported by JSPS Open Partnership Joint Research
Projects Grants JPJSBP120209921 and JPJSBP120203202. T.H. is supported by JSPS Grant-in-Aid for Young Scientists 19K14546; K.N. is supported by JSPS Grant-in-Aid for Young Scientists 21K13807 and received
research environment support from CIMAT as a JSPS Overseas Research
Fellow; N.S. is supported by JSPS Grant-in-Aid for Scientific Research (C)JP19K03515 and JP23K03133; Y.U. is supported by JSPS Grant-in-Aid forYoung Scientists 22K13925. Moreover, the last two authors are supported
by JSPS Grant-in-Aid for Scientific Research (B) 19H01791.

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Takahiro Hasebe
Kei Noba
Department of Statistical Inference and Mathematics
Department of Mathematics
Hokkaido University
The Institute of Statistical Mathematics
Sapporo, 060-0810 Hokkaido, Japan
Tachikawa, 190-8562 Tokyo, Japan
E-mail: thasebe@math.sci.hokudai.ac.jp
E-mail: knoba@ism.ac.jp
Noriyoshi Sakuma
Graduate School of Natural Sciences
Nagoya City University
Nagoya, 467-8501 Aichi, Japan
E-mail: sakuma@nsc.nagoya.ac.jp

Yuki Ueda
Department of Mathematics
Hokkaido University of Education
Asahikawa, 070-8621 Hokkaido, Japan
E-mail: ueda.yuki@a.hokkyodai.ac.jp