ALEA, Lat. Am. J. Probab. Math. Stat. 20, 941–971 (2023)
DOI: 10.30757/ALEA.v20-34

On freely quasi-infinitely divisible distributions
Ikkei Hotta, Wojciech Młotkowski, Noriyoshi Sakuma and Yuki Ueda
Department of Applied Science, Yamaguchi University 2-16-1 Tokiwadai, Ube 755-8611, Japan.
E-mail address: ihotta@yamaguchi-u.ac.jp

Instytut Matematyczny, Uniwersytet Wrocławski, Plac Grunwaldzki 2, 50-384 Wrocław, Poland.
E-mail address: mlotkow@math.uni.wroc.pl

Graduate School of Sciences, Nagoya City University, 1 Yamanohata, Mizuho-cho, Mizuho-ku, Nagoya, Aichi
467-8501, Japan.
E-mail address: sakuma@nsc.nagoya-cu.ac.jp

Department of Mathematics, Hokkaido University of Education, 9 Hokumon-cho, Asahikawa 070-8621,
Japan.
E-mail address: ueda.yuki@a.hokkyodai.ac.jp

Abstract. Inspired by the notion of quasi-infinite divisibility (QID), we introduce and study theclass of freely quasi-infinitely divisible (FQID) distributions on R, i.e. distributions which admit thefree Lévy-Khintchine-type representation with signed Lévy measure. We prove several propertiesof the FQID class, some of them in contrast to those of the QID class. For example, a FQID
distribution may have negative Gaussian component, and the total mass of its signed Lévy measure
may be negative. Finally, we extend the Bercovici-Pata bijection, providing a characteristic triplet,with the Lévy measure having nonzero negative part, which is at the same time classical and freecharacteristic triplet.

1. Introduction
In classical and free probability theories, infinitely divisible distributions are defined by the classical convolution operation and connected with Lévy processes, which are stochastic processes withindependent increments and time-homogenous stationary distributions. More precisely, if µ is an
infinitely divisible distribution on R, then one can construct a Lévy process {Xt }t≥0 such that amarginal law of X1 coincides with µ. Conversely, if a stochastic process {Xt }t≥0 is a Lévy process,then its marginal distribution is infinitely divisible. Because of this relation, infinite divisibility
plays a crucial role in research on the marginal laws of Lévy processes. Readers may consult Sato(1999) for the classical case, and Barndorff-Nielsen and Thorbjørnsen (2002, 2006); Biane (1998)
for the free case.
Received by the editors March 31st, 2022; accepted February 24th, 2023.
2010 Mathematics Subject Classification. 46L54, 60E07.
Key words and phrases. Freely quasi-infinitely divisible distributions, quasi-Lévy measure, Voiculescu transform,
Bercovici-Pata bijection.
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Infinitely divisible distributions are characterized in terms of analytic tools: the characteristic
function in the classical case Sato (1999) and the R-transform or the Voiculescu transform in thefree case Barndorff-Nielsen and Thorbjørnsen (2006); Bercovici and Voiculescu (1993). These representations are called Lévy-Khintchine representations. One of the most beautiful discoveries inthis area is that the Lévy-Khintchine representation in both classical and free probability can beparametrized by a ≥ 0, γ ∈ R, and a Lévy measure ν. The triplet (a, ν, γ) is called characteristic
triplet and free characteristic triplets, respectively. They give a bijection between the classes ofclassical and free infinitely divisible distributions (see Bercovici and Pata (1999, Theorem 1.2)). It
is called the Bercovici-Pata bijection now. The importance of this bijection comes from the studyof the limit theorem in free probability. Indeed, research on free infinitely divisible distributions
has made rapid progress in the last two decades that the rich structure of the class of free infinitely
divisible distributions has been revealed.
Going back to the classical probability case, many researchers have faced similar interesting examples in which the distributions are not infinitely divisible, but have a Lévy-Khintchine-type representation. Namely, in the Lévy-Khintchine representation, the Lévy measure is a signed measure.
A precise definition is given in Section 2. Such a distribution is called a quasi-infinitely divisible
distribution. Although the class of quasi-infinitely divisible distributions has been considered fora long time since being investigated in Linnik and Ostrovskiı̆ (1977), few systematic approaches
for this class have been reported. However, there has been some progress in the past decade. In
particular, Lindner, Pan, and Sato in Lindner et al. (2018) gathered many examples and studied the
distributional properties of quasi-infinitely divisible distributions using the Fourier transform. They
investigate quasi-infinitely divisible distributions from the Lévy-Khintchine representation and thecharacteristic triplet. They also proved in particular, that if (a, ν, γ) is the characteristic triplet ofa quasi-infinitely divisible distribution, then the Gaussian component a must be nonnegative.
The purpose of this paper is to investigate the freely quasi-infinitely divisible (FQID) distributions. The class of FQID distributions is a natural extension of the class of freely infinitely divisible
distributions to explore correspondence between free and classical probability. Section 2 presents
some preliminary results that are used in the remainder of the paper. In Section 3, we define the
FQID distributions and identify several properties (convolution, atoms, convergence, and support)of FQID distributions. In Section 4, we present some examples of FQID distributions that are
not freely infinitely divisible. Moreover, we discover different points about classical and free quasiinfinitely divisible distributions through several examples. In Section 5, we focus on a question
raised by Bożejko of whether the Bercovici-Pata bijection can be extended to a larger class. Finally,
we succeeded in making it from a subclass of QID, which includes ID, to a subclass of FQID, which
includes FID. In order to answer his question, we find examples that come from the Lévy measurewith the Cauchy distribution and symmetric distribution.

2. Preliminaries
2.1. Infinitely divisible distributions. Let B be the set of all Borel sets in R and P(R) the set of all(Borel) probability measures on R. For µ ∈ P(R), its characteristic function is defined byZ
eitx µ(dx),t ∈ R.
µ
b(t) :=R

A probability measure µ ∈ P(R) is said to be infinitely divisible (for short, ID) if, for any n ∈ N,
there exists some µn ∈ P(R) such that µ = µn ∗ · · · ∗ µn = µ∗nn , where ∗ means the (classical){z}|n timesconvolution, that is, for µ, ν ∈ P(R), we define µ ∗ ν as the probability measure on R, characterizedby µ[∗ ν(t) = µb(t)bν (t) for all t ∈ R. The class of ID distributions, denoted by ID(∗), comes fromOn freely quasi-infinitely divisible distributions

943

limit theorems and plays a crucial role in constructing stochastic processes in connection to thestudy of Lévy processes and its applications in mathematical statistics, finance, and physics.
If µ is ID, then its characteristic function µb(t) admits the following representation (namely, theLévy-Khintchine representation):


Z
a 2
itx
µ
b(t) = exp iγt − t + (e − 1 − itx1[−1,1] (x))ν(dx) ,t ∈ R,(2.1)
2
R
where γ ∈ R, a ≥ 0, and ν is a Lévy measure1 on R. Conversely, if a characteristic function ofµ ∈ P(R) is written by the Lévy-Khintchine representation, then µ is ID. Each µ ∈ ID(∗) determines
a unique triplet (a, ν, γ), called a characteristic triplet for µ (see Sato (1999) for further information).
The characteristic function of an ID distribution has another representation. Let µ be an ID
distribution with a characteristic triplet (a, ν, γ). One can see that, if we defineb = γ,
Z(2.2)
ζ(B) = aδ0 (B) + (1 ∧ x2 )ν(dx),B ∈ B,
B

then ζ is a finite measure on R andZµ
b(t) = exp ibt +gc (x, t)ζ(dx) ,t ∈ R,(2.3)

R

where the function gc is defined by((eitx − 1 − itx1[−1,1] (x))/(1 ∧ x2 ), x 6= 0,
gc (x, t) :=−t2 /2,x = 0,t ∈ R.

Observe that the kernel function R → R : x 7→ gc (x, t) is bounded for each t ∈ R and continuous
at 0. It is easy to verify that µ ∈ ID(∗) determines a unique pair (b, ζ), called the characteristic
pair for µ. Conversely, if µ ∈ ID(∗) has a characteristic pair (b, ζ), then µb(t) is written by the
Lévy-Khintchine representation with a characteristic triplet (a, ν, γ) fulfilled the relation (2.2).
2.2. Quasi-infinitely divisible distributions. In this section, we summarize the work of Lindner, Pan,and Sato (Lindner et al., 2018) for quasi-infinitely divisible distributions.
S
A function ρ : B → [−∞, ∞] is called a signed measure on R if ρ(∅) = 0 and ρ( ∞n=1 An ) =
P∞
ρ(A
)
for
any
pairwisedisjoint
sets
{A
}
in
B.
We
definethetotalvariationofa signednn nn=1
measure ρ byX
|ρ|(A) :=sup|ρ(Aj )|.
(2.4)
{Aj }:partition of A j

It is known that |ρ| is a measure on R. A signed measure ρ is said to be finite if |ρ| is a finite
measure. By the Hahn-Jordan decomposition theorem (see e.g., Rudin (1987, p.127)), for a finite
signed measure ρ, there exist disjoint Borel sets C + , C − in R and finite measures ρ+ , ρ− on B
satisfying ρ = ρ+ − ρ− so that ρ+ (R\C + ) = ρ− (R\C − ) = 0. We call ρ+ and ρ− the positive partand the negative part of the signed measure ρ, respectively. The measures ρ+ and ρ− are uniquelydetermined by ρ. It holds that11
ρ− = (|ρ| − ρ)and|ρ| = ρ+ + ρ− .
(2.5)
ρ+ = (|ρ| + ρ),2
2
We now define quasi-infinitely divisible distributions on R.
Definition 2.1. A probability measure µ ∈ P(R) is said to be quasi-infinitely divisible (for short,
QID) if its characteristic function µb(t) admits the representation (2.3) for some b ∈ R and somefinite signed measure ζ on R.
1A positive Borel measure ν on R is called a Lévy measure if ν({0}) = 0 and R (1 ∧ x2 ) ν(dx) < ∞.
R

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Ikkei Hotta et al.

The pair (b, ζ) is uniquely determined by a QID distribution µ and is called a characteristic pair
for a QID distribution µ. The measure µ is QID if and only if there exist µ1 , µ2 ∈ ID(∗) such thatµ ∗ µ1 = µ2 by applying the Hahn-Jordan decomposition to a finite signed measure ζ appeared from
(2.3).
We can rephrase this in terms of the characteristic triplets. We seek a characteristic triplet for
the QID distribution µ in terms of the Lévy-Khintchine representation in the sense of (2.1). We
need to use a little caution here. If there exist µ1 , µ2 ∈ ID(∗) such that µ1 ∗ µ = µ2 and that ν1 andν2 are Lévy measures of µ1 and µ2 satisfying ν1 (R) = ν2 (R) = ∞, respectively, then the differenceν1 − ν2 may not be a signed measure.
To justify the meaning of difference between two Lévy measures, we introduce two subclasses of
B and the class of specific functions as follows.
Definition 2.2 (see Lindner et al. (2018) for details).
S(1) Br := {B ∈ B; B ∩ (−r, r) = ∅} for r > 0 and B0 := r>0 Br .
(2) Denote by Π the set of functions ν : B0 → [−∞, ∞] such that the following condition holds:(C) ν|Br is a finite signed measure on a measurable space (R\(−r, r), Br ) for each r > 0.
The condition (C) ensures that the total variation |ν|, the positive part ν + , and the negative
part ν − are unique and well-defined measures on (R, B) satisfying|ν|({0}) = ν + ({0}) = ν − ({0}) = 0,and|ν|(A) = ν|Br (A), ν + (A) = ν + |Br (A), ν − (A) = ν − |Br (A)for A ∈ Br .

Note that B0 is not a σ-algebra, and so ν ∈ Π is not a signed measure. Whenever we can extend
the definition of ν to B such that ν will be a signed measure, we identify the symbol ν with its
extension to B and speak of ν as a signed measure. In this case, we can verify that ν({0}) = 0 and
its total variation |ν|, positive part ν + and negative part ν − as defined in Definition 2.2 coincidewith the corresponding notions from (2.4) and (2.5) for the signed measure ν.
Definition 2.3. A function ν : B0 → [−∞, ∞] is called a quasi-Lévy-type measure if ν ∈ Π and it
satisfies thatZ
(1 ∧ x2 )|ν|(dx) < ∞.

R

For a QID distribution µ with a characteristic pair (b, ζ), the characteristic function of the QID
distribution µ admits the following representation:


Z
a 2
itx
t ∈ R,µ
b(t) = exp − t + iγt + (e − 1 − itx1[−1,1] (x))ν(dx) ,2Rfor some a, γ ∈ R and some function ν : B0 → [−∞, ∞] defined by the relation (2.2). Note that,the function ν is a quasi-Lévy-type measure. Conversely, if the characteristic function of µ ∈ P(R)admits the above representation for some a, γ ∈ R and a quasi-Lévy-type measure ν on R, then µ is
QID with the characteristic pair (b, τ ) fulfilled the relation (2.2). Note that, b ∈ R and τ is a finite
signed measure on R.
The triplet (a, ν, γ) is uniquely determined by the QID distribution µ and is called a characteristic
triplet of µ.
Definition 2.4. A quasi-Lévy-type measure ν is called a quasi-Lévy measure if there exist a QID
distribution µ and a, γ ∈ R such that (a, ν, γ) is a characteristic triplet of µ.
It is known that a ≥ 0 for any QID distribution with a characteristic triplet (a, ν, γ), see Lindneret al. (2018, Lemma 2.7). Moreover, not every quasi-Lévy-type measure is a quasi-Lévy measure,see Lindner et al. (2018, Example 2.9).

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945

We say that a function f : R → R is integrable with respect to a quasi-Lévy-type measure ν if it
is integrable with respect to |ν| (and therefore with respect to ν + and ν − as well). We then defineZZZ+
f (x)ν − (dx),B ∈ B.
f (x)ν (dx) −f (x)ν(dx) :=B

B

B

Note that, for each t ∈ R, the function x 7→ eitx − 1 − it1[−1,1] (x) is integrable with respect to ν.
2.3. Free probability theory. In non-commutative probability theory, self-adjoint operators are interpreted as (real-valued) random variables. A remarkable feature is that various notions of independence exist for those random variables. In particular, free independence has found applicationsin operator algebras and random matrices and has been intensively studied, see Nica and Speicher(2006); Mingo and Speicher (2017) and references therein.
2.3.1. Free additive convolution. Let X and Y be freely independent random variables affiliatedwith a von Neumann algebra M equipped with a normal faithful tracial state τ . Note that, by
the definition of the affiliated operator, EX (B) and EY (B) belong to M for any Borel set B in
R, where EX is the spectral projection of X. Denote a probability distribution of X with respectto τ as µX , i.e., µX ((−∞, ·]) = τ (EX ((−∞, ·])). A probability distribution of X + Y with respectto τ is denoted by µX  µY with the operation , called free additive convolution. This was
first introduced in Voiculescu (1986), where µ and ν have compact supports. The concept waslater extended to probability measures with finite moments; see Maassen (1992). Finally, it was
generalized by Bercovici and Voiculescu (1993) to the case of probability measures µ and ν on R.
Denote the n-fold free additive convolutionµ  ···  µ|{z}n timesof µ ∈ P(R) as µn .
Free additive convolution is characterized by the Voiculescu transform (see Bercovici and Voiculescu(1993)). A Cauchy-Stieltjes transform Gµ of µ ∈ P(R) is defined byZ
1µ(dx),z ∈ C+ ,Gµ (z) =z−xR
where C+ (resp. C− ) denotes the set of complex numbers with strictly positive (resp. strictly negative) imaginary parts. Note that Im(Gµ (z)) < 0 for any z ∈ C+ . A reciprocal Cauchy transform
Fµ := 1/Gµ is called an F-transform. It is easy to see that Fµ is a Pick function, namely, an analytic
function from C+ to C+ , and Fµ (iy)/iy → 1 as y → ∞. Conversely, a Pick function with a certain
asymptotic behavior at ∞ can be expressed by an F -transform of some probability measure on R.
Proposition 2.5. (see Bercovici and Voiculescu (1993, Proposition 5.2)) If an analytic function
F : C+ → C+ satisfies
F (iy)lim= 1,
y→∞ iythen there exists some µ ∈ P(R) such that F = Fµ .
In classical probability theory, additive convolution ∗ is characterized by the cumulant transform,
that is, the logarithm of the characteristic function: for µ, ν ∈ P(R) we have that log µ[∗ ν(t) =log µb(t) + log νb(t) for all t ∈ R (see e.g., Sato (1999)). In free probability theory, there is a similar
characterization of free additive convolution using the free analog of the cumulant transform. For
any µ ∈ P(R) and λ > 0, there exist positive numbers α, β, and M such that Fµ is univalent on the

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Ikkei Hotta et al.

set Γα,β := {z ∈ C+ | Im(z) > β, |Re(z)| < αIm(z)} and Fµ (Γα,β ) ⊃ Γλ,M . Thus, the compositionalright inverse Fµ−1 is defined on Γλ,M . The Voiculescu transform ϕµ is defined byϕµ (z) := Fµ−1 (z) − z,z ∈ Γλ,M .

It has been shown Bercovici and Voiculescu (1993) that ϕµν (z) = ϕµ (z) + ϕν (z) on the intersectionof the domains where these three Voiculescu transforms ϕµν , ϕµ and ϕν are defined.
2.3.2. Freely infinitely divisible distributions. Since the work of Bercovici and Pata (1999), whofound a bijection between the classical and freely infinitely divisible distributions, the class of
infinitely divisible distributions in free probability has been intensively studied. A probability
measure µ ∈ P(R) is said to be freely infinitely divisible (for short, FID) if, for any n ∈ N, there exists
some µn ∈ P(R) such that µ = µnn . The class of FID distributions is written as ID(), introducedin Bercovici and Voiculescu (1993). Recently, several FID distributions have been discovered: thenormal distributions Belinschi et al. (2011), some of the classical stable laws Hasebe et al. (2020),
some of the gamma distributions, including the chi-square distributions Hasebe (2014), some of theclassical Meixner distributions Bożejko and Hasebe (2013), and of the Fuss-Catalan distributions
Młotkowski et al. (2020); see e.g. Arizmendi and Hasebe (2016); Hasebe (2016); Morishita andUeda (2020) for further examples.
FID distributions can be characterized by those Voiculescu transforms via complex analysis.
Bercovici and Voiculescu (1993) gave the following important result on FID distributions:
Proposition 2.6. For µ ∈ P(R), the following conditions are equivalent.
(i) µ is in ID().
(ii) The Voiculescu transform ϕµ has an analytic extension (denoted by the same symbol ϕµ )defined on C+ with values in C− ∪ R.
(iii) There exist b ∈ R and a finite positive measure τ on R such thatZ
1 + xzτ (dx),z ∈ C+ .
(2.6)
ϕµ (z) = b +R z−x
The pair (b, τ ) is uniquely determined by µ. Conversely, given b ∈ R and a finite positive
measure τ on R, there exists µ ∈ ID() whose Voiculescu transform admits the form (2.6).
The pair (b, τ ) is called the free characteristic pair (or free generating pair, see Barndorff-Nielsen
and Thorbjørnsen (2002, Definition 2.9)) for the FID distribution µ.
The FID distribution is also characterized by admitting the Lévy-Khintchine representation interms of the R-transform. The R-transform (or free cumulant transform) Rµ of µ ∈ P(R) is definedby 
1Rµ (z) = zϕµ, for all z ∈ C− such that 1/z ∈ Γλ,M .
z
Note that ω 7→ 1/ω maps Γλ,M to ∆λ,1/M , where∆α,β := {z ∈ C− : |Re(z)| < −αIm(z), |z| < β},α, β > 0.

(2.7)

Thus, the R-transform Rµ is defined on some subset in ∆λ,1/M .
Remark 2.7. This Rµ (z) is what Nica and Speicher (2006) refer to as the (combinatorial) Rtransform. It is identical to the free cumulant transform Cµ (z) in Barndorff-Nielsen and Thorb
jørnsen (2002, 2006). The formula for the original Voiculesu’s R-transform Rµ is Rµ (z) = ϕµ z1 .
We will employ this combinatorial R-transform to think about Lévy-Khintchine type representation
for both classical and free QID.

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The free version of the Lévy-Khintchine representation amounts to the statement that µ ∈ ID()if and only if there exist a ≥ 0, γ ∈ R and a Lévy measure ν such that

Z 
1
2
Rµ (z) = az + γz +− 1 − zx1[−1,1] (x) ν(dx),z ∈ C− .
(2.8)
1
−
zx
R
The triplet (a, ν, γ), referred to as the free characteristic triplet for µ, is uniquely determined byµ ∈ ID(). The positive number a is called the semicircular component of µ and the measure νis called the free Lévy measure for µ. Furthermore, the representation in (2.8) is called the freeLévy-Khintchine representation of an FID distribution µ. The relation between the free characteristic triplet (a, ν, γ) and free characteristic pair (b, τ ) for the FID distribution µ is as follows (seeBarndorff-Nielsen and Thorbjørnsen (2006)):
a = τ ({0}),1 + x2· 1R\{0} (x) τ (dx),
2
Z x
1 
x 1[−1,1] (x) −γ =b+
ν(dx).
1 + x2R

ν(dx) =

(2.9)

Let us consider µ ∈ P(R). Then, a discrete semigroup {µn }n∈N can be embedded in a continuous
family {µt }t≥1 of probability measures on R, so that µ1 = µ and µt µs = µt+s for t, s ≥ 1. Initially,
Bercovici and Voiculescu (1995) showed the existence of {µt } for sufficiently large t and for µ ∈ P(R)having a compact support. Later, Nica and Speicher (1996) improved their result to t ≥ 1, but µ
still required a compact support. Finally, Belinschi and Bercovici (2004) extended this further to a
general probability measure.
This property can be expressed in the following way: for any µ ∈ P(R) and t ≥ 1, there existsµt ∈ P(R) such that Rµt (z) = tRµ (z) on the intersection of the domains of Rµt and Rµ . We
denote the above probability measure µt as µt . Furthermore, Bercovici and Voiculescu (1993)
showed that a partial semigroup {µt }t≥1 can be embedded in the continuous semigroup {µt }t≥0 ,so that µ0 = δ0 , µ1 = µ, and µt  µs = µt+s for t, s ≥ 0 when µ ∈ ID(). This satisfies
Rµt (z) = tRµ (z) for all z ∈ C− .
2.3.3. The Bercovici-Pata bijection. Let Λ : ID(∗) → ID() be the Bercovici-Pata bijection (seeBercovici and Pata (1999, Theorem 1.2) and Barndorff-Nielsen and Thorbjørnsen (2006) for details).
This bijection satisfies
Λ(µ ∗ ν) = Λ(µ)  Λ(ν)
for all µ, ν ∈ ID(∗), and is continuous with respect to the weak convergence. Moreover, if µ ∈ ID(∗)
has a characteristic triplet (a, ν, γ), then Λ(µ) coincides with the FID distribution with a freecharacteristic triplet (a, ν, γ). For example, we obtain(1) Λ(N(m, σ 2 )) = S(m, σ 2 ) for m ∈ R and σ > 0;(2) Λ(Po(λ)) = MP(λ) for λ > 0;(3) Λ(Ca ) = Ca for a > 0,
where N(m, σ 2 ) is the normal distribution with mean m ∈ R and variance σ 2 , S(m, σ 2 ) is thesemicircle law with mean m ∈ R and variance σ 2 , Po(λ) is the Poisson distribution with a parameter
λ > 0, MP(λ) is the Marchenko-Pastur law with a paramater λ > 0 and Ca is the symmetric Cauchy
distribution with parameter a > 0.
3. Freely quasi-infinitely divisible distributions
3.1. Definition and convolution properties. In this section, we introduce the class of freely quasiinfinitely divisible distributions. Recall that, a measure µ ∈ P(R) is FID if and only if its Voiculescu

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transform ϕµ has an analytic extension to C+ with values in C− ∪ R (see Proposition 2.6). Next,
we will define an extended concept of class ID().
Definition 3.1. A probability measure µ ∈ P(R) is said to be freely quasi-infinitely divisible (forshort, FQID) if ϕµ extends as an analytic function on C+ , and can be expressed as
Z
1 + xzϕµ (z) = b +τ (dx),z ∈ C+ ,z−xR
for some b ∈ R and a finite signed measure τ on R. We call (b, τ ) a free characteristic pair of the
FQID distribution µ.
One immediate consequence of the definition is the following property.
Proposition 3.2. If µ is FQID, then the Voiculescu transform ϕµ has an analytic continuation toC+ .
Example 3.3. Consider µ = 21 (δ−1 + δ1 ). A simple computation gives√z2 − 1−z + z 2 + 4andϕµ (z) =,Fµ (z) =z
2√√
1 i arg(ω)
where ω := |ω| 2 e 2 for 0 ≤ arg(ω) < 2π. The function z 7→ z 2 + 4 has a zero at z = 2i ∈ C+ ,and therefore ϕµ does not have an analytic continuation to C+ . Hence µ is not FQID by Proposition
3.2. In Remark 3.9, we will give a general result on atoms of FQID distributions.
Below, we state and prove some basic properties of FQID distributions.
Proposition 3.4. For an FQID distribution, the free characteristic pair is uniquely determined.
Proof : Let µ be an FQID distribution on R, and let (b1 , τ1 ) and (b2 , τ2 ) be free characteristic pairsof µ. Let (τ1+ , τ1− ) and (τ2+ , τ2− ) be pairs of finite nonnegative measures given by the Hahn-Jordan
decomposition of τ1 and τ2 , respectively. Then, we have thatZ
Z
1 + xz +1 + xz −b1 +τ1 (dx) −τ1 (dx)
R z−x
R z−x
Z
Z
1 + xz +1 + xz −= ϕµ (z) = b2 +z ∈ C+ .
τ2 (dt) −τ2 (dt),
z
−
x
z
−
x
R
R
Therefore, we haveZZ
1 + xz +1 + xz −−(b1 − b2 ) +(τ1 + τ2 )(dt) =(τ1 + τ2+ )(dt),z ∈ C+ .
R z−x
R z−x
Both sides correspond to the Voiculescu transform of some FID probability measure ν. Hence, thepairs (b1 − b2 , τ1+ + τ2− ) and (0, τ1− + τ2+ ) are free characteristic pairs of the FID distribution ν.
By the uniqueness of free characteristic pairs of FID distributions (see Bercovici and Voiculescu(1993)), we have that b1 − b2 = 0 and τ1+ + τ2− = τ1− + τ2+ . Therefore, b1 = b2 and τ1 = τ2 .
Let cB = {cx | x ∈ B}. If ρ is a Borel measure on R and c is a nonzero real constant, then the
dilation of ρ by c is the measure Dc (ρ) given by
Dc (ρ)(B) = ρ(c−1 B),
for any Borel set B.
Proposition 3.5. We have the following properties.
(i) µ is FQID if and only if there exist µ1 , µ2 ∈ ID() such that µ1  µ = µ2 .
(ii) If µ is FQID and µt exists for some t > 0, then µt is also FQID.
(iii) If µ, ν are FQID, c 6= 0 and a ∈ R, then so are µ  ν, Dc (µ) and µ  δa .
(iv) If µ is FID and µ  ν is FQID, then ν is also FQID.

On freely quasi-infinitely divisible distributions

949(v) If µ is FQID and ν is not FQID, then µ  ν is not FQID.
Proof : (i) is obvious from the Hahn-Jordan decomposition theorem. (ii) is a consequence of (i).
We prove (iii). By (i), there exist µ1 , µ2 , ν1 , ν2 ∈ ID() such that µ1  µ = µ2 and ν1  ν = ν2 .
Therefore,(µ1  ν1 )  (µ  ν) = (µ1  µ)  (ν1  ν) = µ2  ν2 .
Since µ1 ν1 , µ2 ν2 ∈ ID(), the measure µν is FQID. Using (i) again, we have that µ1 µ = µ2 .
Hence,Dc (µ1 )  Dc (µ) = Dc (µ1  µ) = Dc (µ2 ).
Because Dc (µ1 ), Dc (µ2 ) ∈ ID(), the measure Dc (µ) is FQID. Furthermore, it follows from (i) thatµ  µ1 = µ2 , and so(µ  δa )  µ1 = δa  (µ  µ1 ) = δa  µ2 .
Since δa  µ1 , δa  µ2 ∈ ID(), the measure µ  δa is also FQID.
Next, we show (iv). As µ  ν is FQID, there exist µ1 , µ2 ∈ ID() such that(µ1  µ)  ν = µ1  (µ  ν) = µ2 .
Because µ1  µ is FID, we conclude that ν is FQID.
Finally, we show (v). Assume that µ  ν is FQID. We then obtain σ1 , σ2 ∈ ID() such that(µ  ν)  σ1 = σ2 by (i). Moreover, there exist µ1 , µ2 ∈ ID() such that µ  µ1 = µ2 , because µ is
also FQID. Then,σ2  µ1 = σ1  (µ  ν)  µ1
= σ1  (µ  µ1 )  ν = (σ1  µ2 )  ν.
Since σ2  µ1 , σ1  µ2 ∈ ID(), the measure ν is FQID, which is a contradiction.

Definition 3.6. For µ1 , µ2 , µ ∈ P(R), such that µ1  µ = µ2 we will write µ = µ2 µ1 . The
operation is called free deconvolution (see Arizmendi et al. (2020) for more information).
Using the operation , we obtain a similar property to that of Proposition 3.5(i), whereby µ is
FQID if and only if µ = µ2 µ1 for some FID distributions µ1 , µ2 .
3.2. Atoms of FQID distributions. In this section, we give a general result on atoms of FQID
distributions.
Theorem 3.7. An FQID distribution has at most one atom.
Theorem 3.7 can be shown by following the same argument as in Bercovici and Voiculescu (1993,
Proposition 5.12(iii)), because the Voiculescu transform of an FQID distribution has an analytic
continuation to C+ . Nevertheless, we include a proof for readers’ convenience.
First, we need the following lemma.
Lemma 3.8. Let µ be an FQID distribution. Then,(i) We have that Fµ (z) + ϕµ (Fµ (z)) = z for all z ∈ C+ .
(ii) Let Ωµ be the component of {z ∈ C+ : Im(z + ϕµ (z)) > 0} that contains iy for some large
number y > 0. Then, Fµ (C+ ) = Ωµ .
Proof : (i) Note that ϕµ (z) = Fµ−1 (z) − z for all z ∈ Γλ,M for some λ, M > 0. Therefore,
ϕµ (Fµ (z)) = z − Fµ (z),z ∈ Fµ−1 (Γλ,M ) ⊂ C+ .

Since ϕµ has an analytic continuation to C+ (denoted by the same symbol ϕµ ), we obtain
Fµ (z) + ϕµ (Fµ (z)) = z,z ∈ C+

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Ikkei Hotta et al.

by the identity theorem.
(ii) Take w ∈ Fµ (C+ ) and z ∈ C+ such that w = Fµ (z). Then,w + ϕµ (w) = Fµ (z) + ϕµ (Fµ (z)) = z ∈ C+ ,and therefore Im(w + ϕµ (w)) > 0. Thus, Fµ (C+ ) ⊂ Ωµ .
Recall that Fµ (z+ϕµ (z)) = z for all z ∈ Fµ (C+ ). The identity extends by analytic continuation toΩµ . Finally, for all z ∈ Ωµ , we have that z +ϕµ (z) ∈ C+ , and therefore z = Fµ (z +ϕµ (z)) ∈ Fµ (C+ ).
This means that Ωµ ⊂ Fµ (C+ ).

Proof of Theorem 3.7: Assume that µ has an atom a ∈ R with the mass β = µ({a}) > 0. We obtainZ−yβ|Gµ (a + iy)| ≥ |ImGµ (a + iy)| =µ(dt) ≥ .
22y
R (a − t) + y
We then have that
|ReFµ (a + iy)| ≤ |Fµ (a + iy)| ≤ImFµ (a + iy)
y
≤
,
β
β

where the last inequality holds by Bercovici and Voiculescu (1993, Corollary 5.3). Therefore, Fµ (a+iy) ∈ Γ1/β,0 for all y > 0. Hence, the curve C := Fµ (a + i(0, ∞)) approaches zero nontangentially.
Moreover, if we define uµ (z) := z + ϕµ (z), the function u maps C to C+ , because C ⊂ Fµ (C+ ) = Ωµholds by Lemma 3.8(ii). Finally, Lemma 3.8(i) implies thatlim

z→0, z∈Cuµ (z) = lim (Fµ (a + iy) + ϕ(Fµ (a + iy))) = lim (a + iy) = a.
y→0

y→0

A consequence of Bercovici and Voiculescu (1993, Lemma 5.11) is that the uniqueness of a nontangential limit a of uµ at the origin holds. Hence the desired result is obtained.

Remark 3.9. In Example 3.3, it was shown that 21 (δ−1 + δ1 ) is not FQID by using the functional
property of its Voiculescu transform. In general, for any a, b ∈ R with a 6= b and any p ∈ (0, 1),
Theorem 3.7 implies that the Bernoulli distribution pδa + (1 − p)δb is not FQID. However, pδa +(1 − p)δb is QID for p 6= 1/2 (see Lindner et al. (2018)). Therefore, quasi-infinite divisibility does
not exhibit complete similarity between the classical and free probability theories.
3.3. Convergence of FQID distributions. We say that a sequence {µn }n ⊂ P(R) converges weakly
to µ ∈ P(R) if, for all f ∈ Cb (R),ZZlimf (x)µn (dx) =f (x)µ(dx),
(3.1)
n→∞ R

R
w

where Cb (R) is the set of all bounded continuous functions on R. In this case, we write µn −→ µ.
The weak convergence of probability measures can be characterized by Voiculescu transforms, see
Bercovici and Voiculescu (1993); Bercovici and Pata (1996).
Lemma 3.10. Let {µn }n ⊂ P(R). Then, the following conditions are equivalent.
w(i) There exists some µ ∈ P(R) such that µn −→ µ as n → ∞.
(ii) There exist positive numbers λ and M and a function ϕ such that all functions ϕµn and ϕ are
defined on Γλ,M , and such that(a) limn→∞ ϕµn (z) = ϕ(z) uniformly on compact subsets of Γλ,M ;ϕµn (z)→ 0 as |z| → ∞ with z ∈ Γλ,M .
(b) supn∈Nz(iii) There exist positive numbers λ and M such that all functions ϕµn are defined on Γλ,M , andsuch that(a) limn→∞ ϕµn (iy) exists for all y ≥ M ;On freely quasi-infinitely divisible distributions951

ϕµn (iy)→ 0 as y → ∞.
yIn this case, we have ϕ = ϕµ on Γλ,M .
(b) supn∈N

A sequence {µn }n of finite (Borel) measures is said to be uniformly bounded if
sup µn (R) < ∞.

(3.2)

n∈N

A sequence {µn }n of finite measures is said to be tight if, for any  > 0, there exists a positive
number T > 0 such thatsup µn (R \ [−T, T ]) ≤ .

(3.3)

n∈N

The above three notions relating to weak convergence, uniform boundedness, and tightness arealso defined for a sequence of finite signed measures on R. If {µn }n is a sequence of finite signed
measures on R, then we say that µn converges weakly to some finite signed measure µ if the measuresµn and µ satisfy condition (3.1), and the family {µn }n is said to be uniformly bounded and tight if
its total variation |µn | satisfies conditions (3.2) and (3.3), respectively.
The following characterization of uniform boundedness and tightness is useful.
Lemma 3.11. We have the following properties.
(i) Let {µn }n be a sequence of finite (signed) measures. Then, the sequence {µn }n is tight if andonly if every subsequence of {µn }n has a weakly convergent subsequence.
(ii) A weakly convergent sequence of finite (signed) measures is uniformly bounded and tight (seeBogachev (2007, Theorem 8.6.2)).
The class ID() is closed with respect to the weak topology. Furthermore, the (weak) convergenceof FID distributions is equivalent to that of free characteristic pairs.
Lemma 3.12. We have the following properties.
(i) The class ID() is weakly closed, that is, if {µn }n is a sequence of FID distributions, µ ∈ P(R),w
and µn −→ µ as n → ∞, then µ is also FID.
(ii) Let {µn }n be a sequence of FID distributions µn with the free characteristic pair (bn , τn ). The
following conditions are equivalent.
w(a) There exists some µ ∈ P(R) such that µn −→ µ as n → ∞;w(b) There exist b ∈ R and a finite measure τ on R such that bn → b and τn −→ τ as n → ∞.
In this case, µ is an FID distribution with the free characteristic pair (b, τ ).
The following theorem extends the implication (b) ⇒ (a) in Lemma 3.12(ii) to FQID distributions.
The proof is a simple modification of Barndorff-Nielsen and Thorbjørnsen (2006, Theorem 5.13(ii)⇒ (i)) to |τn |. However, we include the proof for readers’ convenience.
Theorem 3.13. Let {µn }n be a sequence of FQID distributions with free characteristic pairs (bn , τn ).
w
If there exist b ∈ R and a finite signed measure τ on R such that bn → b and τn −→ τ , then therew
exists some µ ∈ P(R) such that µn −→ µ as n → ∞. Moreover, µ is an FQID distribution with thefree characteristic pair (b, τ ).
Proof : We verify Lemma 3.10(iii) to prove this theorem. As µn is FQID, its Voiculescu transform
ϕµn is defined on C+ . For any y > 0, the map t 7→ 1+tiy
iy−t is continuous and bounded with respect
to t ∈ R. Hence, the convergence of γn and the weak convergence of τn imply thatZZ
1 + tiy
1 + tiyn→∞ϕµn (iy) = γn +τn (dt) −−−→ γ +τ (dt),
y > 0.
iy−t
R
R iy − t
Therefore, limn→∞ ϕµn (iy) exists for all y > 0. Condition (a) in Lemma 3.10(iii) holds.

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For any n ∈ N and y > 0, we have thatZ
ϕµn (iy)
1 + tiyγn=+τn (dt).
yy
R y(iy − t)
Since {γn }n is a convergent sequence, it is bounded; therefore, it suffices to show thatZ
1 + tiysupτn (dt) → 0,y→∞n∈N R y(iy − t)to check condition (b) in Lemma 3.10(iii). For any t ∈ R and y > 0, we have that1
1 + tiy
|t|
≤
+ 2
.
1/2
2
2
y(iy − t)y(y + t )(y + t2 )1/2
For any y ≥ 1, we also have
1 + tiy≤ 2.
t∈R y(iy − t)supFor any N ∈ N and y ≥ 1, we have that

1 + tiyN +1.
≤yt∈[−N,N ] y(iy − t)supTherefore, for any N ∈ N and y ≥ 1, we obtainZZ
1 + tiy
1 + tiysupτn (dt) ≤ sup|τn |(dt)
y(iy
−
t)
y(iy
− t)
n∈N R
n∈N R
N +1
≤
sup |τn |([−N, N ]) + 2 sup |τn |(R \ [−N, N ])
y n∈N
n∈N
N +1
≤
sup |τn |(R) + 2 sup |τn |(R \ [−N, N ]).
y n∈N
n∈N
w

As τn −→ τ , the sequence {|τn |}n is uniformly bounded and tight by Lemma 3.11(ii). Consider anarbitrary  > 0. Since {|τn |}n is tight, there exists some N ∈ N such that supn∈N |τn |(R\[−N, N ]) </4. As supn∈N |τn |(R) < ∞, for any N ∈ N, there is some y0 ≥ 1 such thatN +1

sup |τn |(R) < ,y n∈N2y ≥ y0 .

Therefore, if y ≥ y0 , thenZ
N +1
1 + tiyτn (dt) ≤sup |τn |(R) + 2 sup |τn |(R \ [−N, N ]) < .
sup
y n∈N
n∈N R y(iy − t)n∈Nw

Finally, Lemma 3.10(iii) holds. Therefore, there is a probability measure µ on R such that µn −→µas n → ∞, andZ
1 + tzϕµ (z) = b +τ (dt),z ∈ C+ .
R z−t
Hence, µ is an FQID distribution with the free characteristic pair (b, τ ).

Remark 3.14. The assumptions in Theorem 3.13 cannot be disregarded. To see this, we refer
to Theorem 4.1: let {an }n and {λn }n be sequences of positive numbers such that an ≥ 2 and0 < λn ≤ 1. According to Theorem 4.1, a FQID µn distribution exists when a = an , λ = λn , anddt
c = 1. Its free characteristic pair (bn , τn ) = (− λ2n , π(tan2 +1)− λ2n δ1 ), and its Voiculescu transform is
ϕµn (z) = −an i −

λn z
,
z−1

z ∈ C+ .

On freely quasi-infinitely divisible distributions

953

Thus, it is not FID. If either an → ∞ holds or limn→∞ λn does not exist (e.g. λn = | sin n|), theneither bn or τn (or both of bn and τn ) does not fulfill the assumption in Theorem 3.13. In bothcases, it is easy to see that limz→∞ ϕµn (iy) does not exist for any y > 0. Hence, by Lemma 3.10,µn does not converge weakly to any probability measure.
Remark 3.15. Whether the converse implication of Theorem 3.13 is true remains an open question.
w→ µ, then bn → b and τn → τ as n → ∞. According
More precisely, we wish to know whether, if µn −to the proof in Barndorff-Nielsen and Thorbjørnsen (2006), if µn is an FID distribution with thefree characteristic pair (bn , τn ), then the positivity of τn can be used to show the implication (i) ⇒(ii) in Lemma 3.12(ii). However, if µn is FQID, then τn is a signed measure, and we cannot apply
the techniques used in Barndorff-Nielsen and Thorbjørnsen (2006).
One can obtain the converse of Theorem 3.13 under some additional assumptions regarding freequasi-generating pairs.
Theorem 3.16. Let {µn }n be a sequence of FQID distributions µn with free characteristic pairs(bn , τn ) and let µ be an FQID distribution with the free characteristic pair (b, τ ). Suppose thatwτn = τn+ − τn− for each n ∈ N, where τn± are the positive and negative parts of τ . If µn −→ µ asw
n → ∞, and {τn− }n is tight and uniformly bounded, then bn → b and τn −→ τ as n → ∞.
To prove this theorem, we investigate convergence, tightness, and uniform boundedness for freecharacteristic pairs. Note that the proofs of Lemma 3.17 and Theorem 3.16 are obtained by the
direct application of the results in Lindner et al. (2018).
Lemma 3.17. Let {µn }n be a sequence of FQID distributions with free characteristic pairs (bn , τn ).
If the sequence {µn }n is tight and the sequence {τn− }n is tight and uniformly bounded, then thesequence {bn }n is bounded and the sequences {τn+ }n and {|τn |}n are tight and uniformly bounded.
Proof : As the sequence {µn }n is tight and {τn− }n is tight and uniformly bounded, for any subsequences {kn }n in N, the subsequence {µkn }n is also tight and the subsequence {τk−n }n is also tightand uniformly bounded. By Lemma 3.11(i), there exists a further subsequence {lkn }n of the sequence {kn }n such that the subsequences {µlkn }n and {τl−k }n converge weakly to some probabilitynmeasure µ and some finite measure τ − , respectively.
Let ρlkn be an FID distribution with the free characteristic pair (0, τl−k ) for each n. By Lemmanw

3.12, there is an FID distribution ρ with the free characteristic pair (0, τ − ) such that ρlkn −→ ρ asw
n → ∞. Thus, we have that µlkn  ρlkn −→ µ  ρ as n → ∞. Since µlkn  ρlkn is FID with the free+characteristic pair (blkn , τlk ) and Lemma 3.12(i) holds, the measure µ  ρ is also FID. Therefore,nthe sequence {τl+k }n converges weakly and the sequence {blkn }n converges as n → ∞ by Lemman3.12(ii).
To summarize the above discussion, for any subsequence {kn }n in N, there is a further subsequence{lkn }n of the sequence {kn }n such that the measure τl+k converges weakly and the sequence {blkn }nnconverges. Consequently, the sequence {bn }n is bounded and the sequence {τn+ }n is tight anduniformly bounded; therefore, so is the sequence {|τn |}n .

Proof of Theorem 3.16: By our assumption, the sequence {µn }n is tight, and therefore the sequence{bn }n is bounded and the sequence {τn+ }n is tight and uniformly bounded by Lemma 3.17. Thus,{τn }n is also tight and uniformly bounded. If either (or both) of the sequences {bn }n and {τn }ndoes not satisfy the statement of this theorem, then the tightness and (uniform) boundedness ofthese sequences imply that the sequence {µn }n does not converge weakly, which contradicts the
weak convergence of the sequence {µn }n . Hence, there exist b∗ ∈ R and a finite signed measurewτ ∗ such that bn → b∗ and τn −→ τ ∗ as n → ∞. By the uniqueness of free characteristic pairs (seew
Proposition 3.4) and the convergence µn −→ µ, we have that b = b∗ and τ = τ ∗ .
954

Ikkei Hotta et al.

We mention that the weak convergence preserves the free quasi-infinite divisibility of probability
measures under some additional assumption.
Proposition 3.18. Let {µn }n be a sequence of FQID distributions. For each n ∈ N, let us setµ1,n , µ2,n ∈ ID() satisfying µ1,n  µn = µ2,n . Assume that there exists µ ∈ P(R) such thatw
w
→ µ2 , then there exists→ µ. If there exists µ2 ∈ P(R) (actually, µ2 ∈ ID()) such that µ2,n −µn −w→ µ1 and µ is FQID satisfying µ1  µ = µ2 .
µ1 ∈ P(R) (actually also, µ1 ∈ ID()) such that µ1,n −Proof : Due to Lemma 3.10, there exist M > 0 such that the limits lim ϕµn (iy) and lim ϕµ2,n (iy)
n→∞

n→∞

exist for all y ≥ M , and therefore so is lim ϕ1,n (iy). By the triangle inequality, we haven→∞

ϕµ1,n (iy)ϕµ2,n (iy)ϕµn (iy)
sup
≤ sup
+ sup
→ 0,
y
y
y
n∈N
n∈N
n∈N
as n → ∞. Hence µ1,n satisfies the assumptions in Lemma 3.10(a), (b). Thus there exists µ1 ∈ P(R)w
→ µ1 and µ1 µ = µ2 . Since the class ID() is weakly closed, µ1 is FID, and therefore
such that µ1,n −µ is FQID.

Problem 3.19. In the classical case, the class of all QID distributions is not closed under weak
convergence; see Lindner et al. (2018, Section 4) for a concrete example using the Bernoulli distributions. In the free setting, this example does not work well because the Bernoulli distributions
are not FQID. At the moment, we still do not find an example of a sequence of FQID distributionin which its weak limit is not FQID. Thus, whether the class of all FQIDs is closed under weak
convergence remains an open problem.
3.4. Free Lévy-Khintchine type representations of FQID distributions. In this section, we describe
how R-transforms of FQID distributions give the free Lévy-Khintchine representations.
Lemma 3.20. Let τ be a finite signed measure on R. Then a function ν : B0 → [−∞, ∞] definedbyZ
1 + x2
1R\{0} (x)τ (dx),B ∈ B0 ,ν(B) =
x2
B
is a quasi-Lévy-type measure.
We prove that, for any r > 0, ν|Br is finite signed measure on (R\(−r, r), Br ). It follows from the2
kernel 1+x
is finite on R\(−r, r) and Hahn-Jordan decomposition.
x2
Proof : We fix r > 0. The Hahn-Jordan decomposition implies the existence of a finite measure τ +2is finite on R \ (−r, r), we getand τ − such that τ = τ + − τ − . Since 1+xx2
ZZ
1 + x2
1 + x2+
ν|Br (B) =
1
(x)τ
(dx)
−
1R\{0} (x)τ − (dx),B ∈ Br .
R\{0}
x2
x2
B
B
R
2
1R\{0} (x)τ ± (dx) are finite (positive) measures on Br .
And also, the two functions B 7→ B 1+xx2
Hence ν|Br is a finite signed measure on a measure space (R \ (−r, r), Br ). Moreover, we get
Z
Z
Z
2
2
(1 ∧ x )|ν|(dx) =(1 ∧ x )|ν|(dx) +(1 ∧ x2 )|ν|(dx)(−1,1)\{0}

R\{0}

Z

R\(−1,1)2

x ·=(−1,1)\{0}

1 + x2x2

1 + x2|τ |(dx)x2
R\(−1,1)Z|τ |(dx) +≤ 2 {|τ |((−1, 1) \ {0}) + |τ |(R \ (−1, 1))}≤ 4|τ |(R) < ∞,as desired.

On freely quasi-infinitely divisible distributions

955

Let µ be an FQID distribution with a free characteristic pair (b, τ ). Then the R-transform of µ
is of the form (2.8) for some a, γ ∈ R and some function ν : B0 → [−∞, ∞], fulfilled the relation(2.9). Due to Lemma 3.20, the function ν is a quasi-Lévy-type measure.
Proposition 3.21. Let µ be an FQID distribution with a free characteristic pair (b, τ ). Then, we
have

Z 
1
2
Rµ (z) = az + γz +− 1 − xz1[−1,1] (x) ν(dx),z ∈ C−(3.4)
R 1 − xz
for some a, γ ∈ R and a quasi-Lévy-type measure ν on R. Conversely, if the R-transform of an
FQID distribution µ is of the form (3.4), then µ has a free characteristic pair (b, τ ) fulfilled therelation (2.9).
Moreover, such a triplet (a, ν, γ) is uniquely determined by the FQID distribution µ. The triplet(a, ν, γ) is called a free characteristic triplet of the FQID distribution µ.
Definition 3.22. A quasi-Lévy-type measure ν is called a free quasi-Lévy measure if there exist an
FQID distribution µ and some a, γ ∈ R such that a triplet (a, ν, γ) is a free characteristic triplet ofµ.
Note that, the kernel function of the R-transform of an FQID distribution is integrable withrespect to the quasi-Lévy-type measure.
Proposition 3.23. For z ∈ C− , the function on R defined by1− 1 − zx1[−1,1] (x) =: k(x, z)
x 7→
1 − zxis integrable with respect to a quasi-Lévy-type measure ν.
Proof : Let ν + and ν − be the positive and negative parts of ν, respectively. Then,
Z
Z 1
Z
x2
x
+
2
+
k(x, z)ν (dx) = z
ν (dx) + z
ν + (dx)
R−1 1 − zx
R\[−1,1] 1 − zx
Z 1
Z
1
x
2
2 +
=z
(1 ∧ x )ν (dx) + z
ν + |B1 (dx).
−1 1 − zx
R\[−1,1] 1 − zx
We show that the first term on the right-hand side of the above equality is integrable. Forz = u + iv ∈ C− and x ∈ [−1, 1], we obtain21u2 + v 2
|z| 2
1
= 2
≤
=
.
1 − zx(u + v 2 )x2 − 2ux + 1
v2
Im(z)
R
As R (1 ∧ x2 )ν + (dx) < ∞, we have that
Z 1
Z 1
1
|z|3
2
2 +
z
(1 ∧ x )ν (dx) ≤(1 ∧ x2 )ν + (dx) < ∞,z ∈ C− .
|Im(z)| −1−1 1 − zx
Next, we prove that the second term on the right-hand side of the above equality is also integrable.
For z = u + iv ∈ C− and x ∈ R, we have
2
x
|x|2
1
1
=
≤ 2 =
.
2
2
2
1 − zx(1 − ux) + v xv
Im(z)2As ν + |B1 is finite, we obtainZxzzν + |B1 (dx) ≤ν + |B1 (R \ [−1, 1]) < ∞,
1
−
zx
Im(z)
R\[−1,1]z ∈ C− .

Therefore, the function k(·, z) is integrable with respect to ν + . Similarly, it is also integrablewith respect to ν − , and hence is so with respect to ν.
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Ikkei Hotta et al.

If µ is FQID with a free characteristic triplet (a, ν, η), thenZZ+
2k(x, z)ν − (dx),k(x, z)ν (dx) −Rµ (z) = az + ηz +R

z ∈ C− ,R

by Proposition 3.23. It follows that there exist FID distributions µ+ and µ− with free characteristic
triplets (a, ν + , γ) and (0, ν − , 0), respectively, such that µ−  µ = µ+ .
3.5. FQID distributions with compact support. It is known that, for µ ∈ ID() with a free characteristic pair (b, τ ), the measure µ has a compact support if and only if so does τ , see e.g. Bercovici
and Voiculescu (1992). The purpose of the section is to investigate whether can extend the above
statement to the class of FQID distributions.
We say that a sequence {sn }∞n=0 of complex numbers grows at most exponentially if there exists
a constant c > 0 such that|sn | 6 cn ,for all n ∈ N.

A finite measure ρ has a compact support if and only if the n-th moment mn (ρ) of ρ exists for alln ∈ N and the sequence {mn (ρ)}∞
n=0 grows at most exponentially (see Nica and Speicher (2006,
Lemma 13.13)).
Theorem 3.24. Let µ be an FQID distribution with the free characteristic pair (b, τ ). If the finite
signed measure τ is compactly supported, then so is the measure µ. Furthermore, we have that
τ (R) ≥ −m2 (τ ).
Proof : By the Hahn-Jordan decomposition theorem, we can find two finite measures τ + and τ −such that τ = τ + − τ − . As τ has a compact support, so do τ + and τ − . Let µ+ and µ− be FID
distributions with free characteristic pairs (b, τ + ) and (0, τ − ), respectively. Then µ = µ+ µ−
and µ+ , µ− are compactly supported. By Nica and Speicher (2006, Lemma 13.13 and Proposition
13.15), there exist c+ , c− > 0, such that |κn (µ± )| ≤ cn± , respectively, where κn (ρ) is the n-th free
cumulant of ρ ∈ P(R). Hence there exists c > 0 such that|κn (µ)| ≤ |κn (µ+ )| + |κn (µ− )| ≤ cn .
Hence {κn (µ)}∞
n=1 grows at most exponentially. Also, by Nica and Speicher (2006, Lemma 13.13
and Proposition 13.15), the measure µ is compactly supported. It is elementary to verify thatκn (µ) = mn−2 (τ ) + mn (τ ) for n ∈ N, where m−1 (τ ) = b. Hence, the desired result is obtained byconsidering the case of n = 2.

According to Lindner et al. (2018, Example 3.11), there exists a compactly supported QID distribution such that its quasi-Lévy measure is not compactly supported. That is, the converse
implication of Theorem 3.24 is not true in classical case. On the other hand, we still do not knowwhether the converse statement of Theorem 3.24 is true or not in free case.
Problem 3.25. Suppose that µ is an FQID distribution with compact support and with the freecharacteristic pair (b, τ ). Is the support of τ necessarily compact?
Note also a related problem:
Problem 3.26. Suppose that µ is an FQID distribution with compact support. Is it true that there
exist FID distributions µ1 , µ2 with compact support such that µ1  µ = µ2 ?
The affirmative answer for the latter would imply that for the former.

On freely quasi-infinitely divisible distributions

957

4. Examples
Let R+ := [0, ∞). In the following, the principal square root of z is defined by√
1 i arg(z)z := |z| 2 e 2 , 0 ≤ arg(z) < 2π,(4.1)

namely, the square root function is defined using the nonnegative real axis R+ as a branch cut.
In this section, various examples of FQID distributions (or free characteristic triplets) and theirdistributional properties are provided. For this purpose, let us recall three important families of
distributions in free probability (see Bercovici and Voiculescu (1993); Nica and Speicher (2006);
Mingo and Speicher (2017) for details).
• Let Ca be the Cauchy distribution with the parameter a > 0, that is,1a1R (x)dx.
2
π x + a2
The special case a = 1, the standard Cauchy distribution C1 , is denoted by C. Note thatCa = Da (C) = Ca for all a > 0. The Cauchy distribution is FID and its Voiculescu
transform is given byZ
1 + ztaϕCa (z) =dt = −ai,z ∈ C+ .
2
R z − t π(1 + t )Ca (dx) :=In this case, the free characteristic triplet is (0, πua 2 du, 0).
• Let MP(λ) be the Marchenko-Pastur law with the parameter (variance) λ > 0. The
Voiculescu transform ϕMP(λ) is given byZ
λ1 + tz λλzϕMP(λ) (z) = +δ1 (dt) =,z ∈ C+ .
2
z
−
t
2
z
−
1
R
The measure MP(λ) is FID and the free characteristic triplet is (0, λδ1 , 0). Note that, forc 6= 0, the free characteristic triplet of Dc (MP(λ)) is (0, λδc , 0) and the Voiculescu transformϕDc (MP(λ)) is given byZ
1 + tz λcλczλ+δc (dt) =.
ϕDc (MP(λ)) (z) = 2
2
c +1
z−c
R z−t c +1
• Let S(m, σ 2 ) be the semicircle law with mean m ∈ R and variance σ 2 > 0, i.e.,
1 p 2
S(m, σ 2 )(dx) :=4σ − (x − m)2 1[m−2σ,m+2σ] (x)dx.
2πσ 2The special case S(0, 1) is simply denoted by S. This is FID and the Voiculescu transform
ϕS(m,σ2 ) has the formZ
1 + zt 2
σ2
ϕS(m,σ2 ) (z) = m +σ δ0 (dt) = m + ,z ∈ C+ .
z−tzRIts free characteristic triplet is (σ 2 , 0, m). Note that Da (S(m, σ 2 )) = S(am, a2 σ 2 ).
4.1. Free deconvolution with the Cauchy distribution. In this section, we construct explicit FQID
distributions from the Cauchy distribution. The support of the free Lévy measure of the Cauchy
distribution is unbounded, whereas the support of the free Lévy measure of compactly supported
FID distributions is bounded. Thus, the difference between the free Lévy measure of the Cauchy
distribution and that of a compactly supported FID distribution becomes a signed measure. We
investigate whether there exist FQID probability measures whose quasi-free Lévy measure is a signed
measure.

958

Ikkei Hotta et al.

This idea comes from the results in Arizmendi et al. (2020). If µ ∈ P(R) has a finite variance σ 2 ,then by Arizmendi et al. (2020, Theorem 1.2), there exists some ρ(µ) ∈ P(R) such thatµ  ρ(µ) = C2√2σ .

(4.2)

Furthermore, if µ is an FID distribution with a compact support, then ρ(µ) is FQID. The probability
measures MP(λ) and S(0, σ 2 ) are examples of such µ, as both distributions have finite variances
and are FID with compact supports. That is,• MP(λ)  ρ(MP(λ)) = C2√2λ and ρ(MP(λ)) is FQID,• S(0, σ 2 )  ρ(S(0, σ 2 )) = C2√2σ and ρ(S(0, σ 2 )) is FQID.
In Sections 4.1.1, 4.1.2, and 4.1.3, we investigate signed measures generated from the Lévy measure of the Cauchy distribution and the Marchenko-Pastur, semicircle, and free Meixner distributions, respectively.
4.1.1. Case of the Marchenko-Pastur law. From now on we will denote MP c, λ) := Dc (MP(λ)).
Let a, λ > 0 and c 6= 0. Consider the analytic function
λcz,z ∈ C+ .
(4.3)
z−c
We will prove that ϕ is the Voiculescu transform of some probability measure, or equivalently,that the function K : C+ → C+ defined byϕ(z) = ϕCa (z) − ϕMP(c,λ) (z) = −ai −

λcz
+ z,
z ∈ C+z−cis the compositional right inverse of an F -transform of some probability measure.
A straightforward calculation givesp
z + c(λ + 1) + ai + q(z)−1
F (z) := K (z) =, z ∈ C+ ,2
with an appropriate branch of the square root, whereK(z) := ϕ(z) + z = −ai −(4.4)

q(z) := (z + c(λ + 1) + ai)2 − 4c(z + ai).
A strategy to prove that F is an F-transform of some probability measure, is use of Proposition 2.5. To carry out the proposition, we will show that F is a Pick function which satisfies
limy→∞ F (iy)/iy = 1. It is easy to verify the asymptotic condition for F . Hence, it suffices to show+thatp q(C )+⊂ C \ R+ (recall that+ the branch cut of the square root is R++ in our setting). Then,q(z) ∈ C is well-defined on C , which yields ImF (z) > 0 for all z ∈ C . Let z = x + iy ∈ C+
(namely x ∈ R and y > 0). We have thatq(x + iy) = (x + c(λ + 1))2 − 4cx − (y + a)2 + 2i(y + a)(x + c(λ − 1)).
Then, q(x + iy) ∈ R if and only if x = c(1 − λ). From the equationq(c(1 − λ) + iy) = 4c2 λ − (y + a)2 ,
a 2
it follows that, if 0 < λ 5 2c
, then q(x+iy) ∈/ R+ , which implies that F (C+ ) ⊂ C+ . We conclude
that, in this case, the function F in (4.4) is a Pick function.
By Proposition 2.5, there exists a probability measure ρa,c.λ on R such that F = Fρa,c,λ . Finally,
we obtain
MP(c, λ)  ρa,c,λ = Ca ,and so ρa,c,λ is FQID for all a, λ > 0 and c 6= 0 with 0 < λ 5


a 2
2c .

On freely quasi-infinitely divisible distributions959

2

We further observe that, if 0 < y < c aλ ,Imϕρa,c,λ (c + iy) = −a +

c2 λ
> 0.
y

Hence, the measure ρa,c,λ is not FID. Some calculations indicate that the nontangential limit ofGρa,c,λ at x exists for any x ∈ R. Thus, ρa,c,λ has no singular parts. We summarize the results.
Theorem 4.1. Let a, λ > 0, c 6= 0 with 0 < λ 5that its Voiculescu transform is
ϕρa,c,λ (z) = −ai −a 22c . Then, there exists some ρa,c,λ ∈ P(R) such
λcz
,
z−c

z ∈ C+ .

It has the following properties:(1) MP(c, λ)  ρa,c,λ = Ca . In other words, ρa,c,λ = Ca MP(c, λ) is FQID, rather than FID.
Moreover, it is absolutely continuous with respect to the Lebesgue measure.
adu(2) Its free characteristic triplet is given by (0, πu2 − λδc , 0).
λλc(3) Its free characteristic pair is given by (− c2 +1 , π(uadu
2 +1) − c2 +1 δc ).
Let us consider three specific cases.
√• If a = 2 2λ and c = 1, then ρ2√2λ,1,λ coincides with the measure ρ(MP(λ)) in (4.2).
• If a = 0, then the function ϕ in (4.3) is not the Voiculescu transform of any probability
measure on R.
• If c = a > 0, then for 0 < λ 5 14 , there exists a probability measure ρa,λ := ρa,a,λ ∈ P(R)λazsuch that ϕρa,λ (z) = −ai − z−aby the above discussion. One can see that Da (ρ1,λ ) = ρa,λ .
Example 4.2. We consider the Cauchy transform of ρ1,λ :Gρ1,λ (z) = 1/Fρ1,λ (z)2pz + (λ + 1) + i + (z + (λ + 1) + i)2 − 4(z + i)pz + (λ + 1) + i − (z + (λ + 1) + i)2 − 4(z + i)=2(z + i)p2
z + (λ + 1)(z − i) + 1 − (z − i) (z + (λ + 1) + i)2 − 4(z + i)=,2(z 2 + 1)=

for z ∈ C+ . By the Stieltjes inversion formula, the probability density function f (x) of ρ1,λ is1
f (x) = − lim Im(Gρ1,λ (x + iy))
π y↓0
1
=
{λ + 1 + xv(x) − u(x)} ,2π(x2 + 1)

x ∈ R,

wherehpi(x + (λ + 1) + i)2 − 4(x + i) ,

x∈R

hp
i
(x + (λ + 1) + i)2 − 4(x + i) ,x ∈ R.

u(x) = Re
andv(x) = Im960

Ikkei Hotta et al.

We give an explicit density function for λ = 1/4 below. Set



2
1
q(x) : = x ++ 1 + i − 4(x + i)
41
1
x ∈ R.
= (4x − 3)2 + (4x − 3)i,
16
2
We then obtain q(x) = reiθ , wherer = |q(x)| =p1|4x − 3| 16x2 − 24x + 73
16

and



8
arcsin √,x ≥ 3/4
16x2 −24x+73
θ = arg(q(x)) =8
2π − arcsin √, x < 3/4.
16x2 −24x+73
Using some calculations, we find that
 
p
√
1
θ
u(x) = Re q(x) = r cos
2



p
1
1
8
2
1/4
= sign(4x − 3) |4x − 3|(16x − 24x + 73) cos
arcsin √42
16x2 − 24x + 73
qpp1
16x2 − 24x + 73 + |4x − 3|= √ sign(4x − 3) |4x − 3|
4 2
and
p
√
v(x) = Im q(x) = r sin



1
θ
2






1p
1
8
2
1/4
=
|4x − 3|(16x − 24x + 73) sin
arcsin √42
16x2 − 24x + 73
qp
1 p
|4x − 3|16x2 − 24 + 73 − |4x − 3|,
= √
4 2
where
(
1,
x≥0
sign(x) =−1, x < 0.
Finally, we obtain the density function of ρ1,1/4 (see Figure 4.1):
1
5f (x) =+ xv(x) − u(x)2π(x2 + 1) 4p√
5 2 + |4x − 3| (xp− (x) − sign (4x − 3) p+ (x))√=,
8 2π(x2 + 1)
whereqp
p± (x) :=16x2 − 24x + 73 ± |4x − 3|,x ∈ R.

x ∈ R,

On freely quasi-infinitely divisible distributions

961
0.5

0.4

0.3

0.2

0.1

-3

-2

-1

0

1

2

3

Figure 4.1. Probability density function of ρ1,1/4
4.1.2. Case of the semicircle law. Next, consider
σ2z+
for a, σ > 0 and z ∈ C . We will show that ϕ is the Voiculescu transform of some probability
measure.
2
Let K(z) := z − ai − σz . Then, a formal computation implies thatpz + ai + (z + ai)2 + 4σ 2
−1
F (z) := K (z) =,z ∈ C+ .
2
p
It is easy to see that F (iy)/iy → 1 as y → ∞. For 2σ ≤ a, (z + ai)2 + 4σ 2 is in C+ for all z ∈ C+ .
In a similar manner as for Example 4.1, there is a probability measure γa,σ2 such that F = Fγa,σ2 ,and therefore ϕ = ϕγa,σ2 for all a, σ > 0 with 2σ ≤ a. By definition, it satisfies S(0, σ 2 )γa,σ2 = Ca ,where γa,σ2 is not FID. Moreover, the measure γa,σ2 is absolutely continuous with respect to theLebesgue measure. We summarize the above facts as follows.
ϕ(z) = ϕCa (z) − ϕS(0,σ2 ) (z) = −ai −Theorem 4.3. Let a, σ > 0 with 2σ ≤ a. Then, there exists some γa,σ ∈ P(R) for which the
Voiculescu transform is
ϕγa,σ2 (z) = −ai −σ2,zz ∈ C+ .

It has the following properties:(1) S(0, σ 2 )  γa,σ2 = Ca . In other words, γa,σ2 = Ca S(0, σ 2 ).
(2) γa,σ2 is FQID, rather than FID. Moreover, it is absolutely continuous with respect to theLebesgue measure.
a √
σ 2
(3) γa,σ2 = γ1,(σ/= γa/σ2 ,1 for a ≥ 1 and σ ≥ 1.
a)2
adu
(4) Its free characteristic triplet is given by (−σ 2 , πu
2 , 0).
adu
2(5) Its free characteristic pair is (0, π(1+u2 ) − σ δ0 (du)).
√In particular, γ2√2σ,σ2 (the case in which a = 2 2σ) coincides with the measure ρ(S(0, σ 2 )) in(4.2).

962

Ikkei Hotta et al.

Remark 4.4.
(1) The FQID distribution γa,σ2 has a semicircular component that takes negative values. However,2
if we consider the case of a = 0, then ϕ(z) = − σz is not the Voiculescu transform of any
probability measure.
(2) In general, for a classical characteristic triplet, the Gaussian component must be nonnegative,adu
see Lindner et al. (2018, Lemma 2.7). Therefore the triplet (−σ 2 , πu
2 , 0) in Theorem 4.3 is nota classical characteristic triplet.
Example 4.5. Consider a = 1 and σ = 1/2. The Cauchy transform of γ1,(1/2)2 is given by


p
2
p
z ∈ C+ .
Gγ1,(1/2)2 (z) == −2 z + i − z 2 + 2zi ,z + i + (z + i)2 + 1
By the Stieltjes inversion formula, the probability density function f (x) of γ1,(1/2)2 is
1
f (x) = − lim Im(Gγ1,(1/2)2 (x + iy))
π y↓0
hp
i
2
=
1 − Imx2 + 2xi
π
√


q p
2 √
2
2
=
2 − |x| x + 4 − x ,x ∈ R.
π
Figure 4.2 shows the shape of the probability density function of γ1,(1/2)2 . Clearly, it is not differentiable at x = 0.
0.8

0.6

0.4

0.2

-3

-2

-1

0

1

2

3

Figure 4.2. Probability density function of γ1,(1/2)2

4.1.3. Case of the free Meixner distribution. We define FMa,b as the free Meixner distribution:p4(1 + b) − (x − a)2
FMa,b (dx) :=1[a−2√1+b,a+2√1+b] (x)dx + 0, 1, or 2 atoms,
2π(bx2 + ax + 1)
for a ∈ R and b ≥ −1. If b = −1, then




1
a
1
a
FMa,−1 =1+ √1− √δ 1 (a−√4+a2 ) +δ 1 (a+√4+a2 )2
2
2
2
1 + a2
1 + a2
is not FQID for all a ∈ R by Remark 3.9. Moreover, it follows from Bożejko and Bryc (2006,
Proposition 2.1) that, for −1 < b < 0,


1
t
√
FMa,b = D |b| FM √a ,−1 ,
t=− .
b
|b|

On freely quasi-infinitely divisible distributions

963

Therefore, in view of Proposition 3.3(ii), FMa,b is not FQID for −1 ≤ b < 0. Note that FMa,b ∈ID() if and only if a ∈ R and b ≥ 0, see Saitoh and Yoshida (2001, Theorem 3.2). Furthermore, thevariance of FM0,b is equal to 1 for all b ≥ 0 (in particular, we have FM0,0 = S). Using Arizmendi
et al. (2020, Theorem 1.2), for any b ≥ 0, there exists an FQID distribution ρ(FM0,b ) (in (4.2))
such that
FM0,b  ρ(FM0,b ) = C2√2 .
We summarize the above result as follows.
Theorem 4.6. For any b ≥ 0, there exists some ρ(FM0,b ) such that FM0,b  ρ(FM0,b ) = C2√2 .
For b ≥ 0, the semicircular component of the measure ρ(FM0,b ) is equal to −1, and so the measureρ(FM0,b ) is not FID. Furthermore, for b > 0, the free quasi-Lévy measure νb of the probability
measure ρ(FM0,b ) is given by
 √

√
(
√ √
4b−x2
1
2
−
2
dx, x ∈ [−2 b, 2 b] \ {0}
2
2b
πx
√
νb (dx) =
√ √
2 2
dx,
x ∈ R \ [−2 b, 2 b].
πx2
In particular, we can state the following:b(1) If b ≥ 1/8, then the density function dν
positive;dx (x) is always√ √(2) If 0 < b < 1/8, then there exists some x ∈ [−2 b, 2 b] \ {0} such that the density functiondνbdx (x) of the measure νb is negative. In this case, we have νb (R) = −∞. For example, we show
the density function of the free quasi-Lévy measure ν1/16 in Figure 4.3;
(3) If b = 0, then the free quasi-Lévy measure of ρ(FM0,b ) coincides with the free Lévy measure of
the Cauchy distribution C2√2 .
6

4

2

-2

1

-1

2

-2

Figure 4.3. Density function of the free quasi-Lévy measure ν1/16
4.2. Free deconvolution of the semicircle law and failure of Cramér’s theorem in free probability. For
t > 0, denote∞Xz 2 − tz 4 ± tz 4
Rt± (z) := z 2 ± tz 4
tn z 2n =.
1 − tz 2
n=0

Observe thatRt+ (z) =

√
√
z2
1
z t
1
−z t
√ + √ ·√ .
= √ ·1 − tz 2
2 t 1−z t 2 t 1+z t

964

Ikkei Hotta et al.

This means that Rt+ (z) is the R-transform of a probability distribution µ+ (t), which can be expressed
as




√
√
1
1
t, √  MP − t, √ .
µ+ (t) = MP
2 t
2 t
In particular, µ+ (t) is FID for all t > 0.
For the function Rt− (z), we apply a result by Bercovici and Voiculescu (1995, Theorem 2), which
implies that if t > 0 is sufficiently small, say 0 < t < t0 , then Rt− (z) is the R-transform ofsome compactly supported probability distribution µ− (t) on R. Then, for 0 < t < t0 , we haveRµ+ (t) (z) + Rµ− (t) = 2z 2 , which means that µ+ (t)  µ− (t) = S(0, 2), and consequently,µ− (t) = S(0, 2)µ+ (t).

In view of Bercovici and Voiculescu (1995, Remark 5), the measure µ− (t) cannot be FID, which isalso a consequence of the fact that the fourth free cumulant of µ− (t) is −t < 0.
This example illustrates the observation of Bercovici and Voiculescu (1995) that the free analog
of Cramér’s decomposition theorem Feller (1971, Theorem XV.8.1) (that the Gaussian distribution
cannot be expressed as the classical convolution of two non-Gaussian distributions) is not true.
4.3. Free deconvolution with the Marchenko-Pastur law.
4.3.1. Case of the semicircle law and the Marchenko-Pastur law. In view of Młotkowski (2021,
Proposition 5.1), for every u, x ∈ R, u 6= 0, there exists a probability distribution µ such thatuz
Rµ (z) = x3 u2 z 2 + (1 − x)3.
1 − uzThis means thatu2 x3 , (1 − x)3 δu , 0
is a free characteristic triplet for the measure µ. If x < 0 then we have
µ = MP u, (1 − x)3
S 0, −x3 u2 ,while for x > 1
µ = S 0, x3 u2
MP u, (x − 1)3 u2 .
If x < 0, then the Gaussian component u2 x3 is negative, and therefore (u2 x3 , (1 − x)3 δu , 0) is nota classical characteristic triplet (see Lindner et al. (2018, Lemma 2.7)). The following lemma, which
was proved within Lindner et al. (2018, Example 2.9), implies that if x > 1 then u2 x3 , (1 − x)3 δu , 0
is not a classical characteristic triplet.
Lemma 4.7. Assume that ν is a quasi-Lévy measure, with the positive and negative part ν + , ν − .
If ν − 6= 0 and if either ν + = 0 or supp ν + is a one-point set, then (a, ν, γ) is not the characteristic
triplet of a QID distribution for any a, γ ∈ R.
4.3.2. Case of two Marchenko-Pastur laws. Assume that u, v ∈ R \ {0}, u < v. It was provenin Młotkowski (2021, Proposition 5.3) that for every x ∈ R there exists a compactly supported
probability measure µ such thatRµ (z) =(u − x)3uz(v − x)3
vz
·
+
·
.
2
2
u (u − v) 1 − uz v (v − u) 1 − vz

Putting(u − x)3(v − x)3
,
b(x)
:=
,
(4.5)
u2 (u − v)
v 2 (v − u)
we have a(x) < 0, b(x) > 0 for x < u, a(x), b(x) > 0 for u < x < v and a(x) > 0, b(x) < 0 for
x > v. Thereforeµ = MP v, b(x)MP u, −a(x)a(x) :=

On freely quasi-infinitely divisible distributions

965

for x < u andµ = MP u, a(x)MP v, −b(x)for x > v. In other words, the free characteristic triplet for µ is (0, ν, 0), whereν := a(x) · δu + b(x) · δv .

(4.6)

Note that if u + v 6= 0 then the sum of weights:
a(x) + b(x) =u2 v 2 − 3uvx2 + (u + v)x3
u2 v 2can be negative.
Again, by Lemma 4.7, if either x < u or x > u, then (0, ν, 0), with ν defined by (4.5), (4.6), is
not a classical characteristic triplet.
4.3.3. Case of several Marchenko-Pastur laws. Suppose that u1 , · · · , un ∈ R \ {0} are distinct andn ≥ 2. In view of Liszewska and Młotkowski (2020, Proposition 3.1 and Corollary 2.7), there is a
probability distribution µ on R such that its R-transform is equal toQ
1 − ni=1 (1 − ui z).
(4.7)
Rµ (z) = Qn
i=1 (1 − ui z)
If we settk := Q

ukn−1,
i6=k (uk − ui )thenRµ (z) =

n
X
k=1

tk ·

uk z
.
1 − uk z(4.8)

In the case of n = 2, if u1 < u2 < 0, then t1 > 0 > t2 ; if u1 < 0 < u2 , then t1 , t2 > 0; and if
0 < u1 < u2 , then t1 < 0 < t2 , see Liszewska and Młotkowski (2020) for details. If n ≥ 3, thenti > 0 and tj < 0 for some 1 ≤ i, j ≤ n, see Liszewska and Młotkowski (2020, Lemma 3.5).
Now, assume that either n = 2, u1 u2 > 0, or n ≥ 3. We observe that µ is FQID. Set
K+ := {k : tk > 0} := {k10 , · · · , kp0 },K− := {k : tk < 0} := {k100 , · · · , kq00 }andµ+ := µk10  · · ·  µkp0 ,µ− := µk100  · · ·  µkq00 ,

where

µk :=

MP(uk , tk )if k ∈ K+ ,MP(uk , −tk ) if k ∈ K− .

Then, by (4.8), we have the following deconvolution of µ:µ−  µ = µ+ ;P
the characteristic triplet for µ is (0, ν, 0), where ν := nk=1 tk · δuk . Note that ν(R) = 1. Indeed,taking the limit |z| → ∞ in (4.7) and (4.8), we find that t1 + · · · + tn = 1. For example, if n = 4
and uk = k, thent1 = −1/6, t2 = 4, t3 = −27/2, t4 = 32/3.
One can check that if n is even and the set {u1 , · · · , un } is symmetric, i.e., uk = −un−k+1 , 1 ≤ k ≤ n,then ν is symmetric: tk = tn−k+1 .

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Ikkei Hotta et al.

4.4. Fuss-Catalan distributions. The Fuss-Catalan numbers are defined by
np + rr,
An (p, r) :=nnp + r
where p, r are real parameters and nx := x(x − 1) . . . (x − n + 1)/n! is the generalized binomial
coefficient. The corresponding generating function is of the form∞X

An (p, r)z n = Bp (z)r ,n=0

where the function Bp (z) satisfies the equation Bp (z) = 1 + zBp (z)p on a neighborhood of z = 0,
see Graham et al. (1994). Consider r > 0. It is known in Młotkowski (2010); Młotkowski et al.
(2013); Młotkowski and Penson (2014); Liu and Pego (2016), that the sequence {An (p, r)}∞n=0 ispositive definite if and only if p ≥ 1 and r ≤ p. Let µ(p, r) denote the corresponding probability
distribution on R. The measure µ(p, r) is called the Fuss-Catalan distribution. Then µ(p, r) is FID
if and only if either 1 ≤ p = r ≤ 2 or r ≤ min{p/2, p − 1}, see Młotkowski (2010); Młotkowski et al.
(2020).
Theorem 4.8. If p > 2 and p/2 < r < p − 1 then µ(p, r) is FQID.
Proof : For p > 1 and r > 0 the sequence An (p, r) admits the following integral representation
Z c(p)An (p, r) =xn Wp,r (x) dx,
0where c(p) := pp (p − 1)1−p and Wp,r (x) is given by

Wp,r (ρp (ϕ)) =(sin((p − 1)ϕ))p−r−1 sin(ϕ) sin(rϕ),π(sin(pϕ))p−r

where(sin(pϕ))p,sin(ϕ)(sin((p − 1)ϕ))p−1
is a decreasing function on the interval (0, π/p), which maps (0, π/p) onto (0, c(p)), see Forresterand Liu (2015). Note that if r > p then Wp,r (x) is bounded on (0, c(p)) and admits positive andnegative values. Putρp (ϕ) :=+Wp,r(x) := max{0, Wp,r (x)}

and

−
Wp,r
(x) := − min{0, Wp,r (x)}.

By Młotkowski (2010, Proposition 4.2), the R-transform of µ(p, r) is
Rµ(p,r) (z) = Bp−r (z)r − 1
Therefore, if p > 2 and p/2 < r < p − 1 thenZ c(p)z2
Rµ(p,r) (z) = rz +Wp−r,r (x) dx
1 − xz0
Z c(p)Z c(p)z2
z2+−= rz +
Wp−r,r(x) dx −Wp−r,r(x) dx
1
−
xz
1
−
xz
0
0
= Rµ+ (z) − Rµ− (z),where µ+ , µ− are FID distributions such that
Z c(p)
Rµ± (z) =0z
W ± (x) dx
1 − xz p,rand µ−  µ(p, r) = µ+ , see Nica and Speicher (2006, Theorem 13.16).

On freely quasi-infinitely divisible distributions

967

4.5. Classical characteristic triplets which are not free triplets. Take µa := (1 − a)δ0 + aδ1 , with
0 < a < 1, a 6= 1/2. Then µa is QID by Lindner et al. (2018, Theorem 3.9). More precisely, if
0 < a < 1/2 then the characteristic triplet for µa is τa = (1, νa , 0), where

m
∞
X
a
1
δm ,
νa = −
m a−1
m=1

while if 1/2 < a < 1 then the characteristic triplet is τa = (0, νa , 0), with∞X
1 a−1 mνa = −
δ−m .
m
a
m=1

Proposition 4.9. If 0 < a < 1, a 6= 1/2, then τa is not a free characteristic triplet.
Proof : The characteristic function of µa is given byµca (t) = 1 − a + a exp(it).
Since
log µ
ca (−it) = log (1 − a + a exp(t))
111= at + (a − a2 )t2 + (a − 3a2 + 2a3 )t3 + (a − 7a2 + 12a3 − 6a4 )t4 + · · · ,2
624
the classical cumulants of µa are
r1 = a,r2 = a − a2 ,r3 = a − 3a2 + 2a3 ,r4 = a − 7a2 + 12a3 − 6a4 , . . . .

If τa was the free characteristic triplet for a probability distribution µea , then rn were the free
cumulants for µea . This, by the moment-cumulant formula (see Nica and Speicher (2006, Proposition 11.4)), would imply, that the moments of µea ares0 = 1,s1 = s2 = s3 = a,s4 = a − a2 + 2a3 − a4 , . . . ,which leads to a contradiction, because det(si+j )2i,j=0 = a3 (a − 1)3 < 0.



5. An extension of the Bercovici-Pata bijection
In the previous section, we gave some examples of free characteristic triplets which are not classicalcharacteristic triplets, and also classical characteristic triplets which are not free characteristic
triplets. Here we are going to give some examples of nonpositive triplets, which are at the sametime classical and free characteristic triplets.
Define Φ as the set of such pairs (c, ν), that c is a positive number and ν is a symmetric Lévy
measure, which satisfiesZ
x2 ∨ |x| ν(dx) < ∞.
(5.1)
RNote that (5.1) guarantees that the second moment m2 (ν) of ν is finite. We define three subsets of
Φ, namely, Φ+ is the set of these (c, ν) ∈ Φ that the measurecdx − ν(dx)πx2
is nonnegative, and Φ∗ (respectively, Φ ) will denote the set of such (c, ν) ∈ Φ thatc0, 2 dx − ν(dx), 0
πx
is a classical (respectively, free) characteristic triplet. For (c, ν) ∈ Φ∗ (resp. Φ ) we denote byµ∗ (c, ν) (resp. µ (c, ν)) the corresponding classical (resp. freely) quasi-infinitely divisible distribution. We are going to show that the sets Φ∗ \ Φ+ , Φ \ Φ+ are nonempty, and, remarkably, that

968

Ikkei Hotta et al.

the set Φ∗ ∩ Φ \ Φ+ is nonempty. This will allow us to extend the Bercovici-Pata bijection, thus
to answer affirmatively a question raised by Bożejko.
Proposition 5.1. If (c, ν) ∈ Φ and if the functionZ +∞(1 − cos tx)ν(dx)
φ(t) := exp −c|t| + 2

(5.2)

0

is convex on t ∈ [0, +∞) then (c, ν) ∈ Φ∗ and φ is the characteristic function of the corresponding
distribution µ∗ (c, ν). In particular, if
2

Z ∞
Z ∞
2
x sin tx ν(dx) ≥ 0(5.3)
x cos tx ν(dx) + −c + 2
200

for all t > 0 then (c, ν) ∈ Φ∗ .
Proof : Since
Z

Z ∞

 c
e − 1 − itx1[−1,1] (x)dx − ν(dx) = −c|t| + 2(1 − cos tx)ν(dx),πx2
R0
the first statement is a consequence of Pólya’s theorem; see Pólya (1949) or Feller (1971, Section XV.3). Denoting the left hand side of (5.3) by A(t) we have φ00 (t) = A(t)φ(t), which implies
the second statement.
pProposition 5.2. If (c, ν) ∈ Φ and c ≥ 8m2 (ν), where m2 (ν) denotes the second moment of ν,then (c, ν) ∈ Φ .
itx

Proof : Since ν is a Lévy measure, there is an FID distribution µν such thatZ 
1Rµν (z) =− 1 − zx1[−1,1] (x) ν(dx)R 1 − zx

Z ∞
1
=2
− 1 ν(dx).
1 − z 2 x2
0
Consider arbitrary numbers α, β > 0. As z → 0 with z ∈ ∆α,β , we get

Z ∞
1
2 2
2
− 1 − z x ν(dx)Rµν (z) = m2 (ν)z + 2
1 − z 2 x2
0
= m2 (ν)z 2 + o(z 2 ).
Therefore, by Benaych-Georges (2006, Theorem 1.3), µν has a finite variance equal to m2 (ν) (notethat Rµ (z) in Benaych-Georges (2006) denotes our z −1 Rµ (z)). Hence there is a probability measureρ(µν ) on R such that µν ρ(µν ) = C√8m (ν) by Arizmendi et al. (2020, Theorem 1.2). This implies,2

that (c, ν) ∈ Φ .

Now we restrict ourselves to a special case.
Proposition 5.3. Assume that ν = p (δ−λ + δλ ), p, λ, c > 0, and put p
2p(4p + 1) if 0 < p ≤ 14 ,h(p) :=√
2p + p
if p > 14 .
If c ≥ λ · h(p) then (c, ν) ∈ Φ∗ .
Proof : Denoting again the left hand side of (5.3) by A(t), and using inequality cos α ≥ 12 sin2 α − 1,
we get in our caseA(t) = 2pλ2 cos λt + (−c + 2pλ sin λt)2≥ (4p + 1)pλ2 sin2 λt − 4pλc sin λt − 2pλ2 + c2 .

On freely quasi-infinitely divisible distributions

969

To conclude it suffices to apply the following elementary observation: if a > 0, |y| ≤ 1 and eitherb2 ≤ 4ad (applied for 0 < p ≤ 1/4) or 2a ≤ |b| ≤ a+d (applied for p > 1/4), then ay 2 +by+d ≥ 0. p√Putting c = 8m2 (ν) = 4λ p and applying Proposition 5.3 we obtainp√
Corollary 5.4. Assume that 0 < 4p ≤ 9, λ > 0, ν = p (δ−λ + δλ ) and c = 8m2 (ν) = 4λ p.
Then (c, ν) ∈ Φ∗ ∩ Φ \ Φ+ . Consequently,
c0, 2 dx − ν(dx), 0
πx
is both a classical and a free characteristic triplet.
Remark 5.5. Let T (∗) and T () denote the set of classical and free characteristic triplets, respectively, and let T denote the set of characteristic triplets corresponding to the (classically or freely)
infinitely divisible distributions. From Corollary 5.4 we see that T
T (∗) ∩ T (). We have also
seen that the set T () \ T (∗) is nonempty (Remark 4.4, subsections 4.3.1 and 4.3.2) and the set
T (∗) \ T () is nonempty (Proposition 4.9).
Corollary 5.4 allows us to extend the Bercovici-Pata bijection. Define two families of distributions:Q(∗) := µ ∗ µ∗ (c, ν) : µ ∈ ID(∗), (c, ν) ∈ Φ∗ ∩ Φ ,
Q() := µ  µ (c, ν) : µ ∈ ID(), (c, ν) ∈ Φ∗ ∩ Φ .
We have ID(∗) Q(∗) and ID() Q(). Note that the set Φ∗ ∩Φ is closed under addition, which
implies that Q(∗) and Q() are closed under the classical and the free convolution respectively. Now
e : Q(∗) → Q() by
we can define an extension of the Bercovici-Pata bijection as Λe (µ ∗ µ∗ (c, ν)) := Λ(µ)  µ (c, ν),Λfor µ ∈ ID(∗), (c, ν) ∈ Φ∗ ∩ Φ . This map is well defined, as the characteristic function and thee is an isomorR-transform characterize the corresponding distribution uniquely. One can see that Λeephism between (Q(∗), ∗) to (Q(), ). The bijectivity is clear by the definition of Λ. Moreover, Λ∗
is a homomorphism relative to ∗ and . Indeed, if ρi = µi ∗ µ (ci , νi ) ∈ Q(∗) for i = 1, 2, thene 1 ∗ ρ2 ) = Λe ((µ1 ∗ µ2 ) ∗ µ∗ (c1 + c2 , ν1 + ν2 ))Λ(ρ= Λ(µ1 ∗ µ2 )  µ (c1 + c2 , ν1 + ν2 )
= (Λ(µ1 )  Λ(µ2 ))  µ (c1 , ν1 )  µ (c2 , ν2 )e 1 )  Λ(ρe 2 ).
= Λ(ρRecall that, the original Bercovici-Pata bijection Λ is weakly continuous. Unfortunately, we still
e is weakly continuous or not.
do not know whether the extension Λe weakly continuous? Further, are the two classes Q(∗) and Q() closed
Problem 5.6. Is the map Λwith respect to weak topology?
Acknowledgements
The authors would also like to thank the referees who gave useful comments to improve thispaper. IH was supported by JSPS KAKENHI Grant Numbers JP20K03632. NS was supported
by JSPS KAKENHI Grant Numbers JP19K03515, JP23K03133, JP19H01791, JPJSBP120209921,
JPJSBP120203202. YU was supported by JSPS KAKENHI Grant Numbers JP22K13925,
JP19H01791, JPJSBP120209921, JPJSBP120203202.

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