Fractional integrals and extensions of selfdecomposability, Lecture Notes in Math. (Springer), 2001

Fractional integrals and extensions of
selfdecomposability
Ken-iti Sato

Abstract After characterizations of the class L of selfdecomposable distributions on
Rd are recalled, the classes K p,α and L p,α with two continuous parameters 0 < p < ∞
and −∞ < α < 2 satisfying K1,0 = L1,0 = L are introduced as extensions of the class
L. They are defined as the classes of distributions of improper stochastic integrals
∫ ∞−
(ρ )
(ρ )
f (s)dXs , where f (s) is an appropriate non-random function and Xs is a
0
d
L´evy process on R with distribution ρ at time 1. The description of the classes is
given by characterization of their L´evy measures, using the notion of monotonicity
of order p based on fractional integrals of measures, and in some cases by addition of
the condition of zero mean or some weaker conditions that are newly introduced—
having weak mean 0 or having weak mean 0 absolutely. The class Ln,0 for a positive
integer n is the class of n times selfdecomposable distributions. Relations among the
classes are studied. The limiting classes as p → ∞ are analyzed. The Thorin class T ,
the Goldie–Steutel–Bondesson class B, and the class L∞ of completely selfdecomposable distributions, which is the closure (with respect to convolution and weak
convergence) of the class S of all stable distributions, appear in this context. Some
subclasses of the class L∞ also appear. The theory of fractional integrals of measures
is built. Many open questions are mentioned.
AMS Subject Classification 2000:
Primary:
60E07, 60H05
Secondary: 26A33, 60G51, 62E10, 62H05
Key words: Class L, Class L∞ , Completely monotone, Fractional integral, Improper stochastic integral, Infinitely divisible, L´evy measure, L´evy process, L´evyKhintchine triplet, Monotone of order p, Multiply selfdecomposable, Radial decomposition, Selfdecomposable, Spherical decomposition, Weak mean

Hachiman-yama 1101-5-103, Tenpaku-ku, Nagoya, 468-0074 Japan, e-mail: ken-iti.sato@
nifty.ne.jp; homepage: />
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Ken-iti Sato

1 Introduction
1.1 Characterizations of selfdecomposable distributions
A distribution µ on Rd is called infinitely divisible if, for each positive integer n,
there is a distribution µn such that

µ = µn ∗ µn ∗ · · · ∗ µn ,
n

where ∗ denotes convolution. The class of infinitely divisible distributions on Rd
is denoted by ID = ID(Rd ). Let Cµ (z), z ∈ Rd , denote the cumulant function of
µ ∈ ID, that is, the unique complex-valued continuous function on Rd with Cµ (0) =
0 such that the characteristic function µ (z) of µ is expressed as µ (z) = eCµ (z) . If
µ ∈ ID, then Cµ (z) is expressed as
1
Cµ (z) = − ⟨z, Aµ z⟩ +
2

∫
Rd

(ei⟨z,x⟩ − 1 − i⟨z, x⟩1{|x|≤1} (x))νµ (dx) + i⟨γµ , z⟩. (1.1)

Here ⟨z, x⟩ is the canonical inner product of z and x in Rd , |x| = ⟨x, x⟩1/2 , 1{|x|≤1}
is the indicator function of the set {|x| ≤ 1}, Aµ is a d × d symmetric nonnegativedefinite matrix, called the Gaussian
covariance matrix of µ , νµ is a measure on Rd
∫
2
satisfying νµ ({0}) = 0 and Rd (|x| ∧ 1)νµ (dx) < ∞, called the L´evy measure of
µ , and γµ is an element of Rd . The triplet (Aµ , νµ , γµ ) is uniquely determined by
µ . Conversely, to any triplet (A, ν , γ ) there corresponds a unique µ ∈ ID such that
A = Aµ , ν = νµ , and γ = γµ . Throughout this article Aµ , νµ , and γµ are used in this
sense.
A distribution µ on Rd is called selfdecomposable if, for any b > 1, there is a
distribution µb such that

µ (z) = µ (b−1 z)µb (z),

z ∈ Rd .

(1.2)

Let L = L(Rd ) denote the class of selfdecomposable distributions on Rd . It is characterized in the following four ways.
(a) A distribution µ on Rd is selfdecomposable if and only if µ ∈ ID and its L´evy
measure νµ has a radial (or polar) decomposition

νµ (B) =

∫
S

λ (d ξ )

∫ ∞
0

1B (rξ )r−1 kξ (r)dr

(1.3)

for Borel sets B in Rd , where λ is a finite measure on the unit sphere S = {ξ ∈
Rd : |ξ | = 1} (if d = 1, then S is a two-point set {1, −1}) and kξ (r) is a nonnegative
function measurable in ξ and decreasing and right-continuous in r. (See Proposition
3.1 for exact formulation of radial decomposition.)


Fractional integrals and extensions of selfdecomposability

3

(b) Let {Zk : k = 1, 2, . . .} be independent random variables on Rd and Yn =
n
∑k=1 Zk . Suppose that there are bn > 0 and γn ∈ Rd for n = 1, 2, . . . such that the
law of bnYn + γn weakly converges to a distribution µ as n → ∞ and that {bn Zk : k =
1, . . . , n; n = 1, 2, . . .} is a null array (that is, for any ε > 0, max1≤k≤n P(|bn Zk | >
ε ) → 0 as n → ∞). Then µ ∈ L. Conversely, any µ ∈ L is obtained in this way.
(ρ )
(c) Given ρ ∈ ID, let {Xt : t ≥ 0} be a L´evy process on Rd (that is, a
stochastic process continuous in probability, starting at 0, with time-homogeneous
independent increments, with cadlag paths) having distribution ρ at time 1. If
∫ ∞− −s (ρ )
∫

is de|x|>1 log |x|ρ (dx) < ∞, then the improper stochastic integral 0 e dXs
finable and its distribution
(∫ ∞−
)
(ρ )
µ =L
e−s dXs
(1.4)
0

is selfdecomposable. Here L (Y ) denotes the distribution (law) of a random element Y . Conversely, any µ ∈ L is obtained in this way. On the other hand, if
∫ ∞− −s (ρ )
∫
is not definable. (See Section 3.4 for
|x|>1 log |x|ρ (dx) = ∞, then 0 e dXs
improper stochastic integrals.)
To see that µ of (1.4) is selfdecomposable, notice that
∫ ∞−
0

(ρ )

e−s dXs

∫ log b

=
0

(ρ )

e−s dXs

∫ ∞−

+
log b

(ρ )

e−s dXs

= I1 + I2 ,

I1 and I2 are independent, and
∫ ∞−

I2 =
(ρ )

0

(ρ )

e− log b−s dXlog b+s = b−1
(ρ )

∫ ∞−
0

(ρ )

e−s dYs ,

where {Ys } is identical in law with {Xs }, and hence µ satisfies (1.2).
(d) Let {Yt : t ≥ 0} be an additive process on Rd , that is, a stochastic process
continuous in probability with independent increments, with cadlag paths, and with
Y0 = 0. If, for some H > 0, it is H-selfsimilar (that is, for any a > 0, the two processes
{Yat : t ≥ 0} and {aH Yt : t ≥ 0} have an identical law), then the distribution µ of Y1
is in L. Conversely, for any µ ∈ L and H > 0, there is a process {Yt : t ≥ 0} satisfying
these conditions and L (Y1 ) = µ .
Historically, selfdecomposable distributions were introduced by L´evy [18] in
1936 and written in his 1937 book [19] under the name “lois-limites”, to characterize the limit distributions in (b). L´evy wrote in [18, 19] that this characterization
problem had been posed by Khintchine, and Khintchine’s book [16] in 1938 called
these distributions “of class L”. The book [9] of Gnedenko and Kolmogorov uses
the same naming. Lo`eve’s book [20] uses the name “selfdecomposable”.
The property (c) gives a characterization of the stationary distribution of an
Ornstein–Uhlenbeck type process (sometimes called an Ornstein–Uhlenbeck process driven by a L´evy process) {Vt : t ≥ 0} defined by


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Ken-iti Sato

Vt = e−t V0 +

∫ t
0

(ρ )

es−t dXs ,

(ρ )

where V0 and {Xt : t ≥ 0} are independent. The stationary Ornstein–Uhlenbeck
type process and the selfsimilar process in the property (d) are connected via the
so-called Lamperti transformation (see [11], [26]). For historical facts concerning
(c) see [33], pp. 54–55.
The proofs of (a)–(d) and many examples of selfdecomposable distributions are
found in Sato’s book [39].
The main purpose of the present article is to give two families of subclasses of
ID, with two continuous parameters, related to L, using improper stochastic integrals
and extending the characterization (c) of L.

1.2 Nested classes of multiply selfdecomposable distributions
If µ ∈ L, then, for any b > 1, the distribution µb in (1.2) is infinitely divisible and
uniquely determined by µ and b. If µ ∈ L and µb ∈ L for all b > 1, then µ is called
twice selfdecomposable. Let n be a positive integer ≥ 3. A distribution µ is called
n times selfdecomposable, if µ ∈ L and if µb is n − 1 times selfdecomposable. Let
L1,0 = L1,0 (Rd ) = L(Rd ) and let Ln,0 = Ln,0 (Rd ) be the class of n times selfdecomposable distributions on Rd . Then we have
ID ⊃ L = L1,0 ⊃ L2,0 ⊃ L3,0 ⊃ · · · .

(1.5)

These classes and the class L∞ (Rd ) in Section 1.4 were introduced by Urbanik [52,
53] and studied by Sato [37] and others. (In [37, 52, 53] the class Ln,0 is written as
Ln−1 , but this notation is inconvenient in this article.)
An n times selfdecomposable distribution is characterized by the property that
µ ∈ ID with L´evy measure νµ having radial decomposition (1.3) in (a) with kξ (r) =

hξ (log r) for some function hξ (y) monotone of order n for each ξ (see Section
1.5 and Proposition 2.11 for the monotonicity of order n). In the propeerty (b),
µ ∈ Ln,0 is characterized by the property that L (Zk ) ∈ Ln−1,0 for k = 1, 2, . . .. In
(c), µ ∈ Ln,0 is characterized by ρ ∈ Ln−1,0 in (1.4). A direct generalization of (1.4)
using exp(−s1/n ) or, equivalently, exp(−(n! s)1/n ) in place of e−s is also possible.
In (d), µ ∈ Ln,0 if and only if, for any H, the corresponding process {Yt : t ≥ 0}
satisfies L (Yt −Ys ) ∈ Ln−1,0 for 0 < s < t. The proofs are given in [12, 25, 33, 37].


Fractional integrals and extensions of selfdecomposability

5

1.3 Continuous-parameter extension of multiple
selfdecomposability
In 1980s Nguyen Van Thu [49, 50, 51] defined a continuous-parameter extension
of Ln,0 , replacing the positive integer n by a real number p > 0. He introduced
fractional times multiple selfdecomposability and used fractional integrals and fractional difference quotients. On one hand he extended the definition of n times selfdecomposability based on (1.2) to fractional times selfdecomposability in the form
of infinite products. On the other hand he extended essentially the formula (1.4) in
the characterization (c), considering its L´evy measure.
Directly using improper stochastic integrals with respect to L´evy processes, we
will define and study the decreasing classes L p,0 for p > 0, which generalize the
nested classes Ln,0 for n = 1, 2, . . .. Thus the results of Thu will be reformulated
as a special case in a family L p,α with two continuous parameters 0 < p < ∞ and
−∞ < α < 2. The definition of L p,α will be given in Section 1.6.

1.4 Stable distributions and class L∞
Let µ be a distribution on Rd . Let 0 < α ≤ 2. We say that µ is strictly α -stable if
µ ∈ ID and, for any t > 0, µ (z)t = µ (t 1/α z), z ∈ Rd . We say that µ is α -stable if
µ ∈ ID and, for any t > 0, there is γt ∈ Rd such that µ (z)t = µ (t 1/α z) exp(i⟨γt , z⟩),

z ∈ Rd . (When µ is a δ -distribution, this terminology is not the same as in Sato [39].)
Let Sα0 = S0α (Rd ) and Sα = Sα (Rd ) be the class of strictly α -stable distributions
on Rd and the class of α -stable distributions on Rd , respectively.
Let S = S(Rd )
∪
d
be the class of stable distributions on R . That is, S = 0<α ≤2 Sα . A distribution
µ ∈ ID is in S2 if and only if νµ = 0, that is, µ is Gaussian. A distribution µ ∈ ID
is in Sα with 0 < α < 2 if and only if Aµ = 0 and νµ has a radial decomposition
(1.3) with kξ (r) = r−α . A distribution µ ∈ Sα with 1 < α ≤ 2 is in Sα0 if and
only if µ has mean 0. A distribution µ ∈ S∫1 is in S01 if and only if νµ has a radial
decomposition (1.3) with kξ (r) = r−1 and S ξ λ (d ξ ) = 0. A distribution µ ∈ Sα
with 0 < α < 1 is in Sα0 if and only if it is driftless in the sense that
∫

Cµ (z) =

S

λ (d ξ )

∫ ∞
0

(ei⟨rξ ,z⟩ − 1)r−α −1 dr,

z ∈ Rd .

Lots of results are accumulated on stable distributions and processes. To mention
one of them, the asymptotic behavior of the density of µ ∈ Sα (Rd ), d ≥ 2, α ∈

(0, 2), sensitively depends on the radial direction and exhibits amazing diversity, as
Watanabe [54] shows.
Let L∞ = L∞ (Rd ) denote the smallest class that is closed under convolution and
weak convergence and contains S(Rd ). This class was introduced by Urbanik [52,
53] and reformulated by Sato [37]. If µ ∈ L∞ , then µ ∈ ID with L´evy measure νµ
being such that


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Ken-iti Sato

νµ (B) =

∫
(0,2)

Γ (d α )

∫
S

λα (d ξ )

∫ ∞
0

1B (rξ )r−α −1 dr

(1.6)

for Borel sets B in Rd , where Γ is a measure on the open interval (0, 2) satisfying
∫
(0,2)

(α −1 + (2 − α )−1 )Γ (d α ) < ∞

(1.7)

and {λα : α ∈ (0, 2)} is a measurable family of probability measures on S. This Γ is
determined by νµ , and λα is determined by νµ up to α of Γ -measure 0. Conversely,
if a measure ν on Rd is expressed by the right-hand side of (1.6) with some Γ and
λα satisfying the conditions above, then, for any A and γ , (A, ν , γ ) is the triplet of
some µ ∈ L∞ .
E = LE (Rd ) for a Borel subset E of the open interval
We will also use the class L∞
∞
(0, 2); this is the class of µ ∈ L∞ whose measure Γ is concentrated on E.
Another characterization of L∞ (Rd ) is that µ ∈ L∞ if and only if µ ∈ L and νµ
has a radial decomposition (1.3) with kξ (r) = hξ (log r) where hξ is a completely
monotone function on R for each ξ . Hence we have
∩

L∞ =

Ln,0 .

(1.8)

n=1,2,...

Thus distributions in L∞ are sometimes called completely selfdecomposable.
Zinger [57] introduced a subclass Pr (r being a positive integer) of the class
L(R); it is defined to be the class of limit distributions µ in (b) of Section 1.1 such
that {L (Zk ) : k = 1, 2, . . .} consists of at most r different distributions on R. It is
known that P1 = S(R) and that µ ∈ P2 if and only if µ is the convolution of at
most two stable distributions. In [57] a beautiful explicit description of the L´evy
measures of distributions in Pr is given and it is shown that a distribution in Pr
with r ≥ 3 is not necessarily the convolution of stable distributions on R. Any µ
in Pr is the convolution of at most r semi-stable distributions of a special form.
However, no other characterization of Pr exists, as far as the author knows.

1.5 Fractional integrals
The key concept to connect the representation of L´evy measures for the class L(Rd )
and that for the class L∞ (Rd ) is monotonicity of order p ∈ (0, ∞). It is defined by
using the notion of fractional integrals or Riemann–Liouville integrals. Let us write

Γp = Γ (p),

c p = 1/Γ (p)

throughout this article. The fractional integral of order p > 0 of a function f (s) on
R in a suitable class is given by


Fractional integrals and extensions of selfdecomposability

∫ ∞

cp

r

7

(s − r) p−1 f (s)ds,

which is the interpolation (1 ≤ p < ∞) and extrapolation (0 < p ≤ 1) of the n times
integration
∫ ∞

∫ ∞
r

dsn

sn

dsn−1 · · ·

∫ ∞
s2

f (s1 )ds1 =

1
(n − 1)!

∫ ∞
r

(s − r)n−1 f (s)ds.

However, we need to use fractional integrals of measures. Our definition is as follows.
Let
R+ = [0, ∞), R◦+ = (0, ∞)
and B(E) for the class of Borel sets in a space E. A measure σ is said to be locally
finite on R [resp. R◦+ ] if σ ([a, b]) < ∞ for all a, b with −∞ < a < b < ∞ [resp.
0 < a < b < ∞]. Let p > 0. For a measure σ on R [resp. R◦+ ], let

σ (E) = c p

∫

∫

dr
E

(r,∞)

(s − r) p−1 σ (ds),

E ∈ B(R) [resp. B(R◦+ )].

(1.9)

Let D(I p ) [resp. D(I+p )] be the class of locally finite measures σ on R [resp. R◦+ ]
such that σ is a locally finite measure on R [resp. R◦+ ]. Define
I p σ (E) = σ (E),

E ∈ B(R) [resp. I+p σ (E) = σ (E), E ∈ B(R◦+ )]

for σ ∈ D(I p ) [resp. D(I+p )]. Thus I p and I+p are mappings from measures to measures on R and R◦+ , respectively. D(I p ) and D(I+p ) are their domains.
We call a [0, ∞]-valued function f (r) on R [resp. R◦+ ] monotone of order p on R
[resp. R◦+ ] if
∫
f (r) = c p

(r,∞)

(s − r) p−1 σ (ds)

(1.10)

with some σ ∈ D(I p ) [resp. D(I+p )]. As will be shown in Example 2.17, functions
monotone of order p ∈ (0, 1) have, in general, quite different properties from functions monotone of order p ∈ [1, ∞). We call f (r) completely monotone on R [resp.
R◦+ ] if it is monotone of order p on R [resp. R◦+ ] for all p > 0. This definition of
complete monotonicity differs from the usual one in that positive constant functions
are not completely monotone. Typical completely monotone functions on R and R◦+
are e−r and r−α (α > 0), respectively.
The properties of fractional integrals of functions are studied in M. Riesz [32],
Ross (ed.) [35], Samko, Kilbas, and Marichev [36], Kamimura [15], and others.
Williamson [56] studied fractional integrals of measures on R◦+ for p ≥ 1 and introduced the concept of p-times monotonicity. But we do not assume any knowledge
of them.
In Sections 2.1–2.3 we build the theory of the fractional integral mappings I p and
p
I+ for p ∈ (0, ∞) from the point of view that they are mappings from measures to
measures. A basic relation is the semigroup property I q I p = I p+q and I+q I+p = I+p+q .

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Ken-iti Sato

An important property that both I p and I+p are one-to-one is proved. The relation
between the theories on R and R◦+ is not extension and restriction. We need both
theories, as will be mentioned at the end of Section 6.2.

1.6 Classes K p,α and L p,α generated by stochastic integral
mappings
The formula (1.4) gives a mapping Φ from ρ ∈ ID(Rd ) to µ ∈ ID(Rd ). Thus
(∫ ∞−
)
(ρ )
Φρ = L
e−s dXs
.
(1.11)
0

The domain of Φ is the class of ρ for which the improper stochastic integral in
(1.11) is definable.
For functions f (s) in a suitable class, we are interested in the mapping Φ f from
ρ ∈ ID to µ ∈ ID defined by
(∫ ∞−
)
(ρ )
.
(1.12)

µ = Φf ρ = L
f (s)dXs
0

The domain D(Φ f ) is the class of ρ for which the improper stochastic integral in
(1.12) is definable. The range is defined by R(Φ f ) = {Φ f ρ : ρ ∈ D(Φ f )}.
Let us consider three families of functions. For 0 < p < ∞ and −∞ < α < ∞ let
∫ 1

g¯ p,α (t) = c p

t

∫ 1

j p,α (t) = c p

∫ ∞

gα (t) =

(1 − u) p−1 u−α −1 du,

0 < t ≤ 1,

(− log u) p−1 u−α −1 du,

t

u−α −1 e−u du,

t

0 < t ≤ 1,

0 < t < ∞,

(1.13)
(1.14)
(1.15)

and a¯ p,α = g¯ p,α (0+), b p,α = j p,α (0+), aα = gα (0+). If α < 0, then a¯ p,α =
Γ−α /Γp−α , b p,α = (−α )−p , and aα = Γ−α . If α ≥ 0, then a¯ p,α = b p,α = aα = ∞. Let
t = f¯p,α (s), l p,α (s), and fα (s) be the inverse functions of s = g¯ p,α (t), j p,α (t), and
gα (t), respectively. When α < 0, extend f¯p,α (s) for s ≥ a¯ p,α , l p,α (s) for s ≥ b p,α ,
and fα (s) for s ≥ aα to be zero. Define

Φ¯ p,α = Φ f¯p,α ,

Λ p,α = Φl p,α ,

Ψα = Φ fα .

Sato [42] studied the mapping Ψα and the mapping Φβ ,α = Φ fβ ,α , −∞ < β < α < ∞,
for the inverse function fβ ,α (s) of the function gβ ,α (t) defined by
∫ 1

gβ ,α (t) = cα −β

t

(1 − u)α −β −1 u−α −1 du,

0 < t ≤ 1.


Fractional integrals and extensions of selfdecomposability

9

To make parametrization more convenient, we use Φ¯ p,α = Φα −p,α . For Φ¯ p,α , Λ p,α ,
and Ψα , the domains will be characterized. In the analysis of the domains, asymptotic behaviors of f¯p,α (s), l p,α (s), and fα (s) for s → ∞ are essential. The behaviors
of f¯p,α (s) and fα (s) are similar, but the behavior of l p,α (s) is different from them.
If α ≥ 2, then D(Φ¯ p,α ) = D(Λ p,α ) = D(Ψα ) = {δ0 }. So we will only consider
−∞ < α < 2. Define
K p,α = K p,α (Rd ) = R(Φ¯ p,α ),

(1.16)

L p,α = L p,α (Rd ) = R(Λ p,α ).

(1.17)

It is clear that g¯1,α (t) = j1,α (t), and hence

Φ¯ 1,α = Λ1,α ,

K1,α = L1,α

for −∞ < α < 2.

(1.18)

Since g¯1,0 (t) = j1,0 (t) = − logt, 0 < t ≤ 1, and f¯1,0 (s) = l1,0 (s) = e−s , s ≥ 0, we
have
Φ¯ 1,0 = Λ1,0 = Φ , K1,0 = L1,0 = L
(1.19)
So K p,α and L p,α give extensions, with two continuous parameters, of the class L of
selfdecomposable distributions. Since l p,0 (s) = exp(−(Γp+1 s)1/p ), s ≥ 0, the class
L p,0 coincides with the class of n times selfdecomposable distributions if p is an
integer n.
The following are some of the new results in this article. For any α and p with
−∞ < α < 2 and p > 0, any µ ∈ K p,α has L´evy measure νµ having a radial decomposition
∫
∫

νµ (B) =

S

λ (d ξ )

∞

0

1B (rξ )r−α −1 kξ (r)dr

(1.20)

with kξ (r) measurable in (ξ , r) and monotone of order p on R◦+ in r, and any µ ∈
L p,α has L´evy measure νµ having a radial decomposition

νµ (B) =

∫
S

λ (d ξ )

∫ ∞
0

1B (rξ )r−α −1 hξ (log r)dr

(1.21)

with hξ (y) measurable in (ξ , y) and monotone of order p on R in y. If −∞ < α <
1, then this property of νµ characterizes K p,α and L p,α . If 1∫ < α < 2, then this
property
of νµ combined with the property of mean 0 (that is, Rd |x|µ (dx) < ∞ and
∫
Rd x µ (dx) = 0) characterizes K p,α and L p,α . We will introduce the notion of weak
mean of infinitely divisible distributions in Section 3.3. If α = 1, then the property
above of νµ and the property of weak mean 0 characterize K p,1 ; the case of L p,1 is
still open. For each fixed α , the classes K p,α and L p,α are strictly decreasing as p
increases and at the limit there appear connections with R(Ψα ) and with the class
L∞ of completely selfdecomposable distributions. Namely, define
K∞,α =

∩
0
K p,α ,

L∞,α =

∩
0
L p,α .

(1.22)


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Ken-iti Sato

It will be proved that
K∞,α = R(Ψα ) for −∞ < α < 2,
L∞,α = L∞ for −∞ < α ≤ 0,
E
L∞,α = L∞
E
L∞,α = L∞
∩ {µ :

∫

Rd

with E = (α , 2) for 0 < α < 1,

xµ (dx) = 0}

with E = (α , 2) for 1 < α < 2.

(1.23)
(1.24)
(1.25)
(1.26)

The case of L∞,1 is open.
Combined with the results in [42], the following will be shown. For any α with
−∞ < α < 2, any µ ∈ R(Ψα ) has L´evy measure νµ satisfying (1.20) in which kξ (r)
is measurable in (ξ , r) and completely monotone on R◦+ in r. If −∞ < α < 1, then
this property of νµ characterizes R(Ψα ). If 1 < α < 2, then this property of νµ and
the property of mean 0 characterize R(Ψα ). If α = 1, then this property of νµ and
the property of weak mean 0 characterize R(Ψ1 ).
We will further establish relations among the classes and among stochastic integral mappings. The transformations of L´evy measures corresponding to Φ f , denoted
by Φ Lf , will be examined, which gives the basis of the analysis of the ranges.
Along with the usual improper stochastic integrals Φ f , we will use absolutely
definable improper stochastic integrals and essentially definable improper stochastic
integrals introduced in [41, 42, 43] (see Section 3.4). The domain D0 (Φ f ) of the
former is a subclass of D(Φ f ) and the domain De (Φ f ) of the latter is a superclass of
D(Φ f ). Corresponding to them the absolute range R0 (Φ f ) and the essential range
0 , K e , L0 , and
Re (Φ f ) are introduced. For f = f¯p,α and f = l p,α they define Kp,
p,α

p,α
α
e
L p,α . These classes not only help to study the classes K p,α and L p,α , but also are
interesting classes themselves.
Rosi´nski’s study [34] of tempered stable processes concerns the L´evy processes
associated with distributions in Re (Ψα ), 0 < α < 2, with Gaussian part zero.

1.7 Remarkable subclasses of ID
We have already mentioned the subclasses L, Ln,0 , S, L∞ , K p,α , and L p,α of ID(Rd ).
Let us give the definitions of T , B, and U.
Let us call V x an elementary Γ -variable [resp. elementary mixed-exponential
variable, elementary compound Poisson variable] on Rd if x is a non-random, nonzero element of Rd and V is a real random variable having Γ -distribution [resp. a
mixture of a finite number of exponential distributions, compound Poisson distribution whose jump size distribution is uniform on the interval [0, a] for some a > 0].
Let T = T (Rd ) [resp. B = B(Rd ), U = U(Rd )] be the smallest class of distributions
on Rd closed under convolution and weak convergence and containing the distributions of all elementary Γ -variables [resp. elementary mixed-exponential variables,
elementary compound Poisson variables] on Rd . We call T the Thorin class, B the
Goldie-Steutel-Bondesson class, and U the Jurek class. It is known that


Fractional integrals and extensions of selfdecomposability

11

T = R(Ψ0 ),

(1.27)
(1.28)

B = R(Ψ−1 ),

U = R(Φ¯ 1,−1 ) = K1,−1 .

(1.29)

See [1, 3, 13]. Concerning B and U, notice that f−1 (s) = − log s, 0 < s ≤ 1, so that
(∫ 1
)
(ρ )
Ψ−1 ρ = ϒ ρ = L
(− log s)dXs
,
0

where ϒ is the mapping introduced by Barndorff-Nielsen and Thorbjørnsen [3], and
that f¯1,−1 (s) = 1 − s, 0 ≤ s ≤ 1, so that
(∫ 1
)
(∫ 1
)
(ρ )
(ρ )
¯
s dXs
Φ1,−1 ρ = L
(1 − s)dXs
=L
,
0

0

which is the mapping of Jurek [13]. Noting (1.23), we see that
T = K∞,0 ,

(1.30)

B = K∞,−1 .

(1.31)

Historically, the class of µ ∈ T (R) on the positive axis was introduced by Thorin
[47, 48] in the naming of generalized Γ -convolutions (GGC), to show that Pareto
and log-normal distributions are infinitely divisible. The class of µ ∈ B(R) on the
positive axis was introduced by Bondesson [4] in the naming of generalized convolutions of mixtures of exponential distributions (g.c.m.e.d), after Goldie showed
the infinite divisibility of mixtures of exponential distributions and Steutel found the
description of their L´evy measures. The present formulation of T (Rd ) and B(Rd ) is
by Barndorff-Nielsen, Maejima, and Sato [1]. The class U was introduced by Jurek
[13] as the class of s-selfdecomposable distributions. Our formulation of U(Rd ) is
new; we can prove its equivalence to the definition of Jurek similarly to the proof of
Theorem F of [1].
See Bodesson [5] and Steutel and van Harn [46] for examples and related classes.
Especially, many examples in T (R) are known. To mention one of them, the distribution of L´evy’s stochastic area of the two-dimensional Brownian motion has
density 1/(π cosh x) and belongs to T (R) with L´evy measure dx/(2 |x sinh x|).

2 Fractional integrals and monotonicity of order p > 0
2.1 Basic properties
For α ∈ R, let Mα∞ (R) [resp. Mα∞ (R◦+ )] be the class of locally finite measures σ on
∫
β
R [resp. R◦+ ] such that (1,∞) rα σ (dr) < ∞. For β ∈ R, let M0 (R◦+ ) be the class of

12

Ken-iti Sato

∫

locally finite measures σ on R◦+ such that (0,1] rβ σ (dr) < ∞. Let ML = ML (Rd )
∫
be the class of measures ν on Rd satisfying ν ({0}) = 0 and Rd (|x|2 ∧ 1)ν (dx) < ∞.
That is, ML (Rd ) is the class of L´evy measures of infinitely divisible distributions
on Rd . The words increase and decrease are used in the non-strict sense.
In Section 1.5 we defined the mappings I p and I+p for p > 0 and the notion of
monotonicity of order p. Let us begin with the following remarks. (i) If f is monotone of order p > 0 on R, then the restriction of f to R◦+ is monotone of order p on
R◦+ . (ii) If f is monotone of order p ≥ 1, then f is finite-valued and decreasing. For
p = 1 this is obvious. For p > 1 this follows from Corollary 2.6 to be given later.
(iii) If f is monotone of order p ∈ (0, 1), then f is finite almost everywhere, but f
possibly takes the infinite value at some point and f is not necessarily decreasing.
See Example 2.17 (a), (b), and (d).
Proposition 2.1. Let p > 0. It holds that
D(I p ) = M∞p−1 (R),

(2.1)

D(I+p ) = M∞p−1 (R◦+ ).

(2.2)

Proof. Let σ be a locally finite measure on R [resp. R◦+ ]. Let −∞ < a < b < ∞

[resp. 0 < a < b < ∞]. Then σ of (1.9) satisfies

σ ([a, b]) = c p

∫ b

∫

dr
(r,∞)

a

∫

= cp

(a,∞)

σ (ds)

∫

= c p+1

(s − r) p−1 σ (ds)

(b,∞)

∫ b∧s

a

(s − r) p−1 dr

((s − a) p − (s − b) p )σ (ds) + c p+1

which is finite if and only if

∫

(1,∞) s

p−1 σ (ds) < ∞,

∫
(a,b]

(s − a) p σ (ds),

since

(s − a) p − (s − b) p = s p ((1 − a/s) p − (1 − b/s) p ) ∼ p(b − a)s p−1
as s → ∞.

⊓
⊔

q
Corollary 2.2. If 0 < q < p, then D(I p ) ⊂ D(I q ) and D(I+p ) ⊂ D(I+
).

Proposition 2.3. Let p > 0. Let α > −1 and β > 0.
(i) Let σ ∈ D(I p ) [resp. D(I+p )]. Then I p σ ∈ Mα∞ (R) [resp. I+p σ ∈ Mα∞ (R◦+ )] if
and only if σ ∈ M∞p+α (R) [resp. M∞p+α (R◦+ )].
(ii) Let σ ∈ D(I+p ). Then I+p σ ∈ Mα0 (R◦+ ) if and only if σ ∈ M0p+α (R◦+ ).
∫
∫
(iii) Let σ ∈ D(I p ). Then (−∞,0) eβ r (I p σ )(dr) < ∞ if and only if (−∞,0) eβ s σ (ds)
< ∞.
Assertion (i) is the right-tail fattening property of I p [resp. tail fattening property
of I+p ]. Assertion (ii) is the head thinning property of I+p .


Fractional integrals and extensions of selfdecomposability

13

Proof. (i) Let σ = I p σ [resp. I+p σ ]. We have
∫ ∞
1

rα σ (dr) = c p

∫ ∞

rα dr

(r,∞)

1

∫

= cp

(1,∞)

∫

= cp

(1,∞)

σ (ds)

0

∫ 1

rα dr

rα (s − r) p−1 dr

∫

(0,∞)

∫

(0,1]

σ (ds)

∫ 1
1/s

uα (1 − u) p−1 du.

p+α σ (ds) < ∞.

(r,∞)

∫

= cp

1

(1,∞) s

0

= cp

∫ s

∫

∫

rα (I+p σ )(dr) = c p

(s − r) p−1 σ (ds)

s p+α σ (ds)

Hence (1,∞) rα σ (dr) < ∞ if and only if
(ii) We have
∫ 1

∫

(s − r) p−1 σ (ds)

∫ 1∧s
0

rα (s − r) p−1 dr

f (s)σ (ds) + c p

∫
(1,∞)

g(s)σ (ds)

where
∫ s

f (s) =

0

∫ 1

g(s) =
0

rα (s − r) p−1 dr

for 0 < s ≤ 1,

rα (s − r) p−1 dr

for s > 1.

Since
f (s) = sα +p

∫ 1
0

and
g(s) = sα +p

∫ 1/s
0

∫

uα (1 − u) p−1 du

uα (1 − u) p−1 du ∼ (α + 1)−1 s p−1 ,

s → ∞,

and since (1,∞) s p−1 σ (ds) < ∞, we obtain the assertion.
(iii) We have
∫ 0
−∞

eβ r (I p σ )(dr) = c p

∫ 0
−∞

∫

= cp

R

eβ r dr

σ (ds)

∫

= cp

(−∞,0]

∫

∫

(r,∞)
s∧0

−∞

(s − r) p−1 σ (ds)

eβ r (s − r) p−1 dr

f (s)σ (ds) + c p

∫
(0,∞)

where
∫ s

f (s) =

−∞

eβ r (s − r) p−1 dr

for s ≤ 0,

g(s)σ (ds),


14

Ken-iti Sato

∫ 0

g(s) =
Notice that

−∞

eβ r (s − r) p−1 dr

f (s) = eβ s

∫ ∞

for s > 0.

e−β u u p−1 du

0

and

Using

g(s) = eβ s

∫ ∞
s

∫

(1,∞) s

p−1 σ (ds) < ∞,

e−β u u p−1 du ∼ β −1 s p−1 ,

s → ∞.
⊓
⊔

we can show the result.

Proposition 2.4. For any p > 0 and q > 0,
I q I p = I p+q

q p
I+
I+ = I+p+q .

and

(2.3)

As always an equality of mappings includes the assertion that the domains of both
hands are equal.
Lemma 2.5. Let p > 0 and q > 0. If σ ∈ D(I p ) [resp. D(I+p )] and σ = I p σ [resp.
I+p σ ], then
∫

cq

(u,∞)

(r − u)q−1 σ (dr) = c p+q

∫
(u,∞)

(s − u) p+q−1 σ (ds)

(2.4)

for u ∈ R [resp. R◦+ ].
Proof. We have
∫
(u,∞)

cq (r − u)q−1 σ (dr) =
∫

= c p cq

(u,∞)

∫

= c p cq

σ (ds)

(u,∞)

u

u

cq (r − u)q−1 dr

(u,∞)

∫
(r,∞)

c p (s − r) p−1 σ (ds)

(r − u)q−1 (s − r) p−1 dr

(s − u) p+q−1 σ (ds)

∫

= c p+q

∫ s

∫ ∞

∫ 1
0

(1 − v)q−1 v p−1 dv

(by change of variables v = (s − r)/(s − u))
(s − u) p+q−1 σ (ds),
⊓
⊔

which is (2.4).

Proof of Proposition 2.4. We prove the first equation in (2.3), but the proof of
the second one is formally the same. The domain of I q I p is defined to be {σ ∈
D(I p ) : I p σ ∈ D(I q )}. It follows from Propositions 2.1 and 2.3 (i) that

σ ∈ D(I q I p )

⇔

σ ∈ M∞p−1 (R),

⇔

σ ∈ M∞p+q−1 (R)

I p σ ∈ Mq−1
∞ (R)


Fractional integrals and extensions of selfdecomposability

15

σ ∈ D(I p+q ).

⇔

If σ ∈ M∞p+q−1 (R), then Lemma 2.5 shows that (I q (I p σ ))(du) = (I p+q σ )(du). ⊓
⊔
Corollary 2.6. Let 0 < q < p. If a function f is monotone of order p on R [resp.
R◦+ ], then f is monotone of order q on R [resp. R◦+ ].

2.2 One-to-one property
We will prove an important result that I p and I+p are one-to-one. We prepare auxiliary
mappings Dq and Dq+ and two lemmas, suggested by Kamimura [15].
Definition 2.7. Let 0 < q < 1. Let D(Dq ) [resp. D(Dq+ )] be the class of locally finite
measures ρ on R [resp. R◦+ ] absolutely continuous with density g(s) such that
∫ ∞
r

(s − r)−q−1 |g(s) − g(r)|ds < ∞

for a. e. r ∈ R [resp. R◦+ ]

(2.5)

and that the signed measure ρ defined by
(
)
∫ ∞
ρ (dr) = qc1−q
(s − r)−q−1 (g(s) − g(r))ds dr

(2.6)

r

has locally finite variation on R [resp. R◦+ ]. Define
Dq ρ = ρ

[resp. Dq+ ρ = ρ ]

(2.7)

for ρ ∈ D(Dq ) [resp. D(Dq+ )].
The reason for introducing the mappings Dq and Dq+ is seen from the following
lemma.
Lemma 2.8. Let 0 < q < p < 1 and let σ ∈ D(I p ) [resp. D(I+p )]. Then I p σ ∈ D(Dq )
p
)] and
[resp. I+p σ ∈ D(D+
(Dq I p σ )(dr) =

Γp−q
(qC p,q − 1) (I p−q σ )(dr)

ΓpΓ1−q

(2.8)

[resp. the same equality with Dq+ , I+p , and I+p−q in place of Dq , I p , and I p−q ], where
∫ 1

C p,q =

0

(1 − u)−q−1 (u p−1 − 1)du.

Proof. Let ρ = I p σ [resp. I+p σ ]. Then ρ (ds) = g(s)ds with g(s) = c p
s) p−1 σ (du). For s > r we have
g(s) − g(r)

(2.9)
∫

(s,∞) (u −


16

Ken-iti Sato

= −c p
= −c p

∫
(r,s]

∫

(r,s]

(u − r) p−1 σ (du) + c p

∫
(s,∞)

((u − s) p−1 − (u − r) p−1 )σ (du)

(u − r) p−1 σ (du) + (1 − p)c p

∫
(s,∞)

σ (du)

∫ s
r

(u − v) p−2 dv.

Let
∫ ∞

J1 =

r

(s − r)−q−1 ds

J2 = (1 − p)
∫ ∞

Then

r

∫ ∞
r

∫
(r,s]

(u − r) p−1 σ (du),

−q−1

(s − r)

∫

σ (du)

ds
(s,∞)

∫ s
r

(u − v) p−2 dv.

(s − r)−q−1 |g(s) − g(r)|ds ≤ c p (J1 + J2 ).

Since σ ∈ D(I p−q ) [resp. D(I+p−q )], we have
∫

J1 =

(r,∞)

= q−1

(u − r)

∫
(r,∞)

J2 = (1 − p)
= (1 − p)
= (1 − p)
= (1 − p)
= (1 − p)
= (1 − p)

σ (du)

∫ ∞
u

(s − r)−q−1 ds
for a. e. r ∈ R [resp. R◦+ ],

(u − r) p−q−1 σ (du) < ∞

∫

(r,∞)

∫

(r,∞)

∫

(r,∞)

∫

(r,∞)

∫

(r,∞)

∫

∫

= C p,q

p−1

(r,∞)

(r,∞)

σ (du)
σ (du)
σ (du)
σ (du)

∫ u
r

∫ u
r

(u − v) p−2 dv
(u − v) p−1 dv

∫ 1

∫ u

dt

0

∫ 1
0

r

t −p dt

0

(u − r) p−q−1 σ (du)

0

∫ 1
0

∫ t(u−r)

(u − r) p−q−1 σ (du)

∫ 1

C p,q =

v

(s − r)−q−1 ds
(u − r − t(u − v))−q−1 dt

(u − v) p−1 (u − r − t(u − v))−q−1 dv

∫ 1
0

∫ 1
0

(u − r) p−q−1 σ (du) < ∞

where

∫ u

w p−1 (u − r − w)−q−1 dw

t −p dt

∫ t
0

x p−1 (1 − x)−q−1 dx

x p−1 (1 − x)−q−1 dx

∫ 1

t −p dt

x

for a. e. r ∈ R [resp. R◦+ ],

x p−1 (1 − x)−q−1 (1 − x1−p )dx = C p,q

and the finiteness of C p,q is clear since (1 − x)−q−1 (1 − x1−p ) ∼ (1 − p)(1 − x)−q as
x ↑ 1. We have thus shown (2.5) and
∫ ∞
r

(s − r)−q−1 (g(s) − g(r))ds = c p (J2 − J1 )


Fractional integrals and extensions of selfdecomposability

= c p (C p,q − q−1 )

∫
(r,∞)

17

(u − r) p−q−1 σ (du).

Hence I p σ ∈ D(Dq ) [resp. I+p σ ∈ D(Dq+ )] and
(
∫
p p
(D I σ )(dr) = c1−q c p (qC p,q − 1)

(u − r)

(r,∞)
p−q

= Γp−q c1−q c p (qC p,q − 1)I

p−q−1

)
σ (du) dr

σ (dr)

on R, and similarly on R◦+

⊓
⊔

Lemma 2.9. Let p > 0 and let σ ∈ D(I p ) [resp. D(I+p )]. Then,
Iqσ → σ

[resp. I+q σ → σ

vaguely on R

vaguely on R◦+ ]

(2.10)

as q ↓ 0, that is, for all continuous functions f with compact support in R [resp. R◦+ ],
∫

f (s)I q σ (ds) →

∫

f (s)σ (ds)

[resp.

∫

q
f (s)I+
σ (ds) →

∫

f (s)σ (ds)]

(2.11)

as q ↓ 0.
Proof. We give the proof in the case R, but the case R◦+ is similar. First recall that
σ ∈ D(I p ) implies σ ∈ D(I q ) for 0 < q ≤ p. Assume that f is nonnegative, continuous with support in [a, b] for some a < b. It is enough to show (2.11) for such f .
Notice that
∫
R

f (s)I q σ (ds) =

∫

∫

R

f (r)dr
(r,∞)

∫ s

where
gq (s) =

−∞

cq (s − r)q−1 σ (ds) =

∫
R

gq (s)σ (ds),

cq (s − r)q−1 f (r)dr.

We claim that
gq (s) → f (s),

q↓0

(2.12)

for s ∈ R. We have gq (s) = 0 = f (s) for s ≤ a. Fix s > a. Let q be such that a <
s − q < s. Then, as q ↓ 0,
|gq (s) − f (s)| ≤ cq

∫ s−q
a

(s − r)q−1 f (r)dr + cq

∫ s

+ cq

s−q

∫ s
s−q

(s − r)q−1 | f (r) − f (s)|dr

(s − r)q−1 dr − 1 f (s)

= J1 + J2 + J3 ,
J1 ≤ cq || f ||

∫ s−q

where || f || = maxs∈R f (s),

a

(s − r)q−1 dr = cq+1 || f ||((s − a)q − qq ) → 0,


18

Ken-iti Sato

J2 ≤ max | f (r) − f (s)|cq+1 qq → 0,
r∈[s−q,s]

J3 = |cq+1 qq − 1| f (s) → 0.
This proves (2.12). If s > a, then
gq (s) ≤ cq || f ||

∫ s
a

(s − r)q−1 dr = cq+1 || f ||(s − a)q ≤ const ((s − a) ∨ 1) p

for 0 < q ≤ p. If s > b + 1, then
gq (s) ≤ cq || f ||

∫ b
a

(s − r)q−1 dr ≤ cq || f ||(b − a)(s − b)q−1 ≤ const (s − b) p−1

for 0 < q ≤ p. Now, since σ ∈ M∞p−1 (R), we can use the dominated convergence
theorem and obtain
∫
R

gq (s)σ (ds) →

∫
R

f (s)σ (ds),

q ↓ 0,
⊓
⊔

completing the proof.
Theorem 2.10. For any p > 0, I p and I+p are one-to-one.

Proof. Assume that p < 1. Suppose that σ1 , σ2 ∈ D(I p ) satisfy I p σ1 = I p σ2 . Let
0 < q < p. By virtue of Lemma 2.8, I p σ j ∈ D(Dq ) for j = 1, 2 and (2.8) holds
for σ = σ1 , σ2 . We have Dq I p σ1 = Dq I p σ2 . If qC p,q − 1 ̸= 0, then it follows that
I p−q σ1 = I p−q σ2 . From the definition (2.9), C p,q is positive and strictly increasing
with respect to q. Hence, either qC p,q − 1 ̸= 0 for all q ∈ (0, p) or there is q0 ∈ (0, p)
such that qCp,q − 1 ̸= 0 for all q ∈ (0, p) \ {q0 }. Thus
qC p,q − 1 ̸= 0
Hence

I p−q σ1 = I p−q σ2

for all q ∈ (0, p) sufficiently close to p.

(2.13)

for all q ∈ (0, p) sufficiently close to p.

Now, letting q ↑ p and using Lemma 2.9, we obtain σ1 = σ2 . It follows that I p is
one-to-one for 0 < p < 1. Now, using Proposition 2.4, we see that I p is one-to-one
if p = np′ with a positive integer n and 0 < p′ < 1. Hence I p is one-to-one for any
p > 0. The proof for I+p is similar.
⊓
⊔

2.3 More properties and examples
When p is a positive integer, we have the following characterization. This is a result
of Williamson [56]. It is given also in Lemmas 3.2 and 3.4 of Sato [37] based on
Widder’s book [55].


Fractional integrals and extensions of selfdecomposability

19

Proposition 2.11. (i) A function f (r) on R [resp. R◦+ ] is monotone of order 1 if and
only if it is decreasing and right-continuous on R [resp. R◦+ ] and tends to 0 as r → ∞.
(ii) Let n be an integer ≥ 2. A function f on R [resp. R◦+ ] is monotone of order n
if and only if



 f (r) tends to 0 as r → ∞ and is n − 2 times differentiable on R
(2.14)
[resp. R◦+ ] with (−1) j f ( j) ≥ 0 for j = 0, 1, . . . , n − 2, and with


n−2
(n−2)
(−1)
f
being decreasing and convex.
Corollary 2.12. Let n be an integer ≥ 1. Suppose that f is n times differentiable on
R [resp. R◦+ ]. Then f is monotone of order n if and only if (−1) j f ( j) ≥ 0 on R [resp.
R◦+ ] for j = 0, 1, . . . , n, and f (r) → 0 as r → ∞.
Thus the concept of complete monotonicity of f on R◦+ coincides with that in
Widder [55] and Feller [8] except the condition that limr→∞ f (r) = 0. Integral representation of a completely monotone function on R◦+ (as the Laplace transform of
a measure on R◦+ ) is obtained from Bernstein’s theorem. A completely monotone
function on R is also represented by the Laplace transform of a measure on R◦+ .
Proof of Proposition 2.11. In this proof we consider the case R. In the case R◦+ ,
replace R by R◦+ .
∫
(i) Recall that f is monotone of order 1 on R if and only if f (r) = (r,∞) σ (ds)
for some σ ∈ M0∞ (R), hence if and only if f (r) is finite, decreasing, and rightcontinuous on R and tends to 0 as r → ∞.
(ii) Let n ≥ 2. A function f is monotone of order n on R if and only if, for some
σ ∈ Mn−1
∞ (R),
∫

∫

1
(s − r)n−1 σ (ds) =
σ (ds)
(r,∞) (n − 1)!
(r,∞)
∫
∫
1
=
(s − u)n−2 σ (ds).
du
(r,∞)
(u,∞) (n − 2)!

∫ s

f (r) =

r

1
(s − u)n−2 du
(n − 2)!

If f is monotone of order n on R, then f (r) → 0 as r → ∞, since f is monotone of
order 1 on R. If f is monotone of order 2 on R, then
∫

f (r) =

(r,∞)

σ ((u, ∞))du

(2.15)

and hence f is decreasing and convex. Conversely, if f (r) is decreasing, convex,
and convergent to 0 as r → ∞, then f is written as in (2.15) with some σ ∈ M1∞ (R)
and hence f is monotone of order 2 on R.
Now let n ≥ 3 and suppose that assertion
(ii) is true with n−1 in place of n. If f is
∫
1
monotone of order n on R, then g(u) = (u,∞) (n−2)!
(s − u)n−2 σ (ds) is monotone of
order n − 1 on R and, a fortiori, continuous and hence − f ′ (r) = g(r), which shows
that (2.14) is satisfied. Conversely, suppose that f satisfies (2.14). Then − f ′ (r) → 0
as r → ∞, since otherwise f (r) goes to −∞ as r → ∞. Hence (2.14) is satisfied with


20

Ken-iti Sato

− f ′ in place of f and with n − 1 in place of n. Hence
∫

1
(s − u)n−2 σ (ds)
(n

−
2)!
(u,∞)

′

− f (u) =

′
for some σ ∈ M∫n−2
∞ (R). Since − f (u) is continuous and since f (r) → 0 as r → ∞,
∞ ′
we have f (r) = r f (u)du and hence

∫

f (r) =

1
(s − r)n−1 σ (ds).
(n
−
1)!
(r,∞)

As (2.14) implies that f is locally integrable on R, σ belongs to D(I n ) and f is
monotone of order n on R.
⊓
⊔
Let us give some necessary conditions for f to be monotone of order p.

Proposition 2.13. Suppose that f is monotone of order p on R [resp. R◦+ ] for some
p > 0 and that f is not identically zero. Then:
(i) f is lower semi-continuous on R [resp. R◦+ ].
(ii) Either f (r) > 0 for all r ∈ R [resp. R◦+ ] or there is a ∈ R [resp. R◦+ ] such that
f (r) > 0 for r < a and f (r) = 0 for r ≥ a.
(iii) In the case of R, lim inf ( f (r)/|r| p−1 ) > 0.
r→−∞

(iv) In the case of R◦+ , lim inf f (r) > 0.
r↓0

Proof. The function f satisfies (1.10) for some σ ∈ M∞p−1 (R) [resp. M∞p−1 (R◦+ )]
with σ ̸= 0.
(i) Using Fatou’s lemma, we see
lim
inf f (r′ ) ≥ c p
′
r →r

= cp

∫
∫

lim
inf(1(r′ ,∞) (s)(s − r′ ) p−1 )σ (ds)
′
r →r

(r,∞)

(s − r) p−1 σ (ds) = f (r),

that is, f is lower semi-continuous.
∫
(ii) If f (r0 ) > 0 for some r0 , then f (r) > 0 for all r ≤ r0 , because (r0 ,∞) (s −
r0 ) p−1 σ (ds) > 0 shows that there is a point s0 in the support of σ such that s0 > r0 .
(iii) Choose −∞ < a < b < ∞ such that σ ((a, b)) > 0. Let r < a. Then
{
∫
c p (b − r) p−1 σ ((a, b))
if p ≤ 1,
p−1
f (r) ≥ c p
(s − r) σ (ds) ≥
p−1
if p > 1.
c p (a − r) σ ((a, b))
(a,b)
Hence the assertion follows.
(iv) Proved similarly to (iii).

⊓
⊔

Proposition 2.14. Suppose that f is monotone of order p on R [resp. R◦+ ] for some
p > 1. Then f is absolutely continuous on R [resp. R◦+ ].


Fractional integrals and extensions of selfdecomposability

21

Proof. Consider the case of R. We have (1.10) for f with some σ ∈ D(I p ). Since
I p = I 1 I p−1 , it follows from Lemma 2.5 that
∫

f (r) =
(r,∞)

(I p−1 σ )(ds) =

∫ ∞

g(s)ds
r

for some g(s) ≥ 0. The case of R◦+ is similar.

⊓
⊔

Let S = Sd−1 = {ξ ∈ Rd : |ξ | = 1}. This is the (d − 1)-dimensional unit sphere in
if d ≥ 2 and the two-point set {−1, 1} if d = 1. A family {σξ : ξ ∈ S} of locally
finite measures on R [resp. R◦+ ] is called a measurable family if σξ (E) is measurable
family,
in ξ ∈ S for every E ∈ B(R) [resp. B(R◦+ )]. If {σξ : ξ ∈ S} is a measurable
∫
then, (a) for any [0, ∞]-valued function f (r, s) measurable in (r, s), f (r, s)σξ (ds) is
measurable in (ξ , r), and (b) for any a > 0, σξ ((r, r + a]) is measurable in (ξ , r). To

see (a), use the monotone class theorem. To see (b), apply (a) to f (r, s) = 1(r,r+a] (s).
Rd

Proposition 2.15. Let p > 0. If {σξ : ξ ∈ S} is a measurable family of measures
in M∞p−1 (R) [resp. M∞p−1 (R◦+ )], then {I p (σξ ) : ξ ∈ S} [resp. {I+p (σξ ) : ξ ∈ S}] is a
measurable family.
Proof. Notice that, for any E ∈ B(R)
I p (σξ )(E) =

∫

(r,∞)

E

∫

=

∫

dr
R

σξ (ds)

c p (s − r) p−1 σξ (ds)

∫
E∩(−∞,s)

c p (s − r) p−1 dr,

which is measurable in ξ . The case of R◦+ is similar.

⊔
⊓
M∞p−1 (R◦+ )].

[resp.
If
Proposition 2.16. Let p > 0 and let {σξ : ξ
p
p
{I (σξ ) : ξ ∈ S} [resp. {I+ (σξ ) : ξ ∈ S}] is a measurable family, then {σξ : ξ ∈ S}
is a measurable family.
∈ S} ⊂ M∞p−1 (R)

Proof. Consider the case of R. The case of R◦+ is similar. Let {I p (σξ )} be a measurable family. For each ξ
I p (σξ )(E) =

∫

∫

E

Let

gξ (r)dr,

∫ r+1/n

gξ (r) = lim inf n
n→∞

r

gξ (r) =

(r,∞)

c p (s − r) p−1 σξ (ds).

gξ (r′ )dr′ = lim inf nI p (σξ )((r, r + 1/n]).
n→∞

Then gξ (r) is measurable in (ξ , r) and, by Lebesgue’s differentiation theorem,
gξ (r) = gξ (r) for a. e. s for every fixed ξ . Thus I p (σξ )(dr) = gξ (r)dr.
Suppose 0 < p < 1. Let 0 < q < p. Then {Dq I p (σξ ) : ξ ∈ S} is a measurable
family. It follows from Lemma 2.8 and (2.13) that {I p−q (σξ ) : ξ ∈ S} is a measurable family for q sufficiently close to p. Hence, by Lemma 2.9, {σξ : ξ ∈ S} is a
measurable family. Now, for any p > 0, write p = np′ with positive integer n and
0 < p′ < 1 and use Proposition 2.4 to see {σξ : ξ ∈ S} is a measurable family. ⊓
⊔


22

Ken-iti Sato

Example 2.17. Let p > 0. In the following, σ is in M∞p−1 (R) or in M∞p−1 (R◦+ ) and
we write
∫
f p (r) = c p
(s − r) p−1 σ (ds)
(2.16)
(r,∞)

∈ R◦+ .

for r ∈ R or for r
Thus f p is monotone of order p on R or on R◦+ .
(a) A δ -distribution located at x is denoted by δx . Let σ = δa with a ∈ R [resp.
R◦+ ]. Then
{
c p (a − r) p−1 ,
r < a,
f p (r) =
0,
r ≥ a.
Hence f p (r) is strictly increasing for r < a if p < 1; f p equals 1 for r < a if p = 1;
f p is not continuous if p ≤ 1. If p > 1, then f p is strictly decreasing for r < a and
continuous on R [resp. R◦+ ]. For any p′ > p, f p is not monotone of order p′ . Indeed,
′
′
otherwise Proposition 2.4 and Theorem 2.10 show that δa = I p −p τ [resp. I+p −p τ ]
′
′
′
for some τ ∈ M∞p −1 (R) [resp. M∞p −1 (R◦+ )], which is absurd since I p −p τ [resp.

′
I+p −p τ ] is absolutely continuous.
Notice that this function f p (r) has the following property: if α ∈ R satisfies
α (p − 1) > −1, then f p (r)α is monotone of order α (p − 1) + 1 and not monotone
of order p′ for any p′ > α (p − 1) + 1.
(b) Let −∞ < a < b < ∞ [resp. 0 < a < b < ∞] and let σ (ds) = 1(a,b] (s)ds. Then

p
p

c p+1 ((b − r) − (a − r) ),
f p (r) = c p+1 (b − r) p ,


0,
Thus

r < a,
a ≤ r < b,
r ≥ b.

f p′ (r) = c p ((a − r) p−1 − (b − r) p−1 ) for r < a.

Hence, if p < 1, then f p is strictly increasing for r ≤ a and strictly decreasing for
a ≤ r ≤ b. For all p > 0, f p is continuous on R [resp. R◦+ ]. For any p′ > p, f p is not
′
′
monotone of order p′ on R [resp. R◦+ ]. Indeed, otherwise σ = I p −p τ [resp. I+p −p τ ]
′
′

for some τ ∈ M∞p −1 (R) [resp. M∞p −1 (R◦+ )], which contradicts Proposition 2.13.
(c) Let σ (ds) = s−α ds on R◦+ with α > p. Then σ ∈ M∞p−1 (R◦+ ) and the function
f p is monotone of order p on R◦+ and
∫ ∞

f p (r) = c p

r

(s − r) p−1 s−α ds = c p r p−α

∫ ∞
1

(u − 1) p−1 u−α du = c+ r p−α

for r > 0, where
∫ ∞

c+ = c p

0

u p−1 (u + 1)−α du = c p B(p, α − p) = Γα −p /Γα .

(d) Suppose 0 < p < 1. Let σ (ds) = (s − b)−α 1(b,∞) (s)ds on R with 1 > α > p
and b ∈ R. Then σ ∈ M∞p−1 (R) and f p is monotone of order p on R and


Fractional integrals and extensions of selfdecomposability


p−α ,

c− (b − r)
f p (r) = ∞,


c+ (r − b) p−α ,

23

rr=b
r > b,

where c+ is the same as in (c) and
∫ ∞

c− = c p

0

(u + 1) p−1 u−α du = c p B(1 − α , α − p) = Γ1−α Γα −p /(ΓpΓ1−p ).
∫

Note that f p (b) = c p b∞ (s − b) p−α −1 ds = ∞. This f p is a (0, ∞]-valued continuous
function on R, strictly increasing on (−∞, b), equal to ∞ at b, and strictly decreasing
on (b, ∞). For any p′ > p, this f p is not monotone of order p′ by the same reason as
in (b).

(e) Let 0 < p < α < 1. Let B = {b1 , b2 , . . .} be a countable set in R. Choose
Cn > 0, n = 1, 2, . . ., satisfying

∑

∑

Cn +

bn ∈B∩(−∞,1]

Let

σ (ds) =

bn ∈B∩(1,∞)

Cn bnp−α < ∞.

∞

∑ Cn (s − bn )−α 1(bn ,∞) (s)ds.

n=1

Then σ

∈ M∞p−1 (R),
∫
(1,∞)

since we have
∞

∑ Cn

s p−1 σ (ds) =

∫ ∞
bn ∨1

n=1

s p−1 (s − bn )−α ds < ∞,

noting that, for bn ≤ 1,
∫ ∞

s
1

p−1

−α

(s − bn )

ds ≤

∫ ∞

s

p−1

1

−α

(s − 1)

∫ ∞

ds =

(u + 1) p−1 u−α du

0

= B(1 − α , α − p)
and, for bn > 1,
∫ ∞
bn

s p−1 (s − bn )−α ds = bnp−α

∫ ∞
1

u p−1 (u − 1)−α du = bnp−α B(1 − α , α − p).

Let f p,α ,b (s) denote the function in (d). Then
∞

f p (r) =

∑ Cn f p,α ,bn (r)

n=1

and f p (bn ) = ∞ for n = 1, 2, . . .. If the set B has supremum ∞, then lim sup f p (s) = ∞.
s→∞

If B is a dense set in R, then f p (s) is finite almost everywhere but infinite on the
dense set.
⊓
⊔


24

Ken-iti Sato

Example 2.18. (a) Let f (r) = r−β for r > 0 with β > 0. Then f is completely monotone on R◦+ , because, for any p > 0, we can choose α = p + β and apply Example
2.17 (c). Alternatively, use Proposition 2.11.
(b) Let f (r) =∫ e−r for r ∈ R. Then f is
completely monotone on R. Use Propo∫
sition 2.11 or c p r∞ (s − r) p−1 e−s ds = c p 0∞ u p−1 e−u−r du = e−r .
(c) Let
{

arcsin(1 − r),
0 < r < 1,
f (r) =
0,
r ≥ 1.
Then f is monotone of order 2 on R◦+ , since it is decreasing and convex. For any
p > 2, f is not monotone of order p on R◦+ . To prove this, suppose f is monotone of
order p > 2 on R◦+ . Then f (r)dr = I+p σ for some σ ∈ M∞p−1 (R◦+ ). Hence f (r)dr =
I+1 τ with τ = I+p−1 σ . On the other hand
∫ ∞

f (r) =

g(s)ds
r

with g(s) = (1 − (1 − s)2 )−1/2 1(0,1) (s).

Hence τ (ds) = g(s)ds by Theorem 2.10. Hence g(s) is equal almost everywhere on
R◦+ to a function monotone of order p − 1. Since p − 1 > 1, it follows that g(s) is
equal almost everywhere on R◦+ to an absolutely continuous function (Proposition
2.14). This is absurd.
(d) Let
{
− log r,
0 < r < 1,
f (r) =
0,
r ≥ 1.
Then, similarly to the previous example, f is monotone of order 2 on R◦+ but is not

monotone of order p on R◦+ for any p > 2.
⊓
⊔
√
Example 2.19. Let g(r) = r2 + 1 − r, r ∈ R, and hα (r) = g(r)α , r ∈ R, with α ∈
(0, ∞). The function g is monotone of order 2 on R, since g(r) > 0, −g′ (r) = 1 −
r(r2 + 1)−1/2 > 0, and
g′′ (r) = (r2 + 1)−1/2 − r2 (r2 + 1)−3/2 = (r2 + 1)−3/2 > 0,
√
g(r) = |r| 1 + |r|−2 − r = |r|(1 + O(|r|−2 )) − r = O(r−1 ), r → ∞.
Let us show the following.
(a) For every α > 0, hα is not monotone of order p on R for any p > α + 1.
(b) For every α > 0, hα is monotone of order 1 on R.
(c) The following statement is true for n = 1, 2, 3. For any α ≥ n, hα is monotone
of order n + 1 on R.
We have g(r) = 2|r| + O(|r|−1 ), r → −∞. Hence we see (a) by virtue of Proposition 2.13 (iii), because hα (r)/|r| p−1 ∼ 2α /|r| p−α −1 as r → −∞. We have (b), since
√
√
α ( r2 + 1 − r)α −1
−α hα
′
√
(r − r2 + 1) = √
,
(2.17)
hα =
r2 + 1
r2 + 1

Fractional integrals and extensions of selfdecomposability

25

which is negative on R. We have (c) for n = 1, since
(
)
√
α hα
rhα
h′α
′′
√
hα = α
−
=
(r
+
α
r2 + 1),
(r2 + 1)3/2
(r2 + 1)3/2
r2 + 1

(2.18)

which is positive on R for α ≥ 1.
The following recursion formula is known for the derivatives of hα ([30] p. 41):
( j+2)

(r2 + 1)hα

( j+1)

+ (2 j + 1)rhα

( j)

+ ( j2 − α 2 )hα = 0.

(2.19)

Indeed, this is true for j = 0 from (2.17) and (2.18); if (2.19) is true for a given
j ≥ 0, then its differentiation shows that it is true with j + 1 in place of j.
Now let us prove (c) for n = 2. It follows from (2.17), (2.18), and (2.19) that
2 ′
′′
(r2 + 1)h′′′
α = −3rhα − (1 − α )hα =

=
=

−α hα
(3α r
2
(r + 1)3/2

√

√
(1 − α 2 )α hα
−3α rhα
(r + α r2 + 1) + √
3/2
2
(r + 1)
r2 + 1

r2 + 1 + (α 2 + 2)r2 + (α 2 − 1))

√
−α hα
3
[
α
(
r2 + 1 + r)2 + (α − 2)(α − 1)r2 + (α − 2)(α + 12 )],
(r2 + 1)3/2 2

which is negative on R for α ≥ 2.
Let us prove (c) for n = 3. We have
(r2 + 1)hα = −5rhα′′′ − (4 − α 2 )h′′α
√
5α rhα
= 2
(3α r r2 + 1 + (α 2 + 2)r2 + (α 2 − 1))
5/2
(r + 1)
√

α hα
− (4 − α 2 ) 2
(r
+
α
r2 + 1)
(r + 1)3/2
√
√
α hα
2
2
2 + 1 + (α 2 − 4)α r 2 + 1
= 2
[(
α
+
11)
α
r
r
(r + 1)5/2
(4)

+ 6(α 2 + 1)r3 + 3(2α 2 − 3)r]
√
α hα
2
3
+

1)(
= 2
[
(
α
r2 + 1 + r)3 + 32 (α 2 − 9)r
(r + 1)5/2 2
√
√
+ (α 3 − 6α 2 + 11α − 6)r2 r2 + 1 + (α 3 − 32 α 2 − 4α − 32 ) r2 + 1 ]
√
√
α hα
[ 23 (α 2 + 1)( r2 + 1 + r)3 + 32 (α 2 − 9)( r2 + 1 + r)
= 2
5/2
(r + 1)
√
√
+ (α − 3)(α − 2)(α − 1)r2 r2 + 1 + (α − 3)(α 2 − 4) r2 + 1 ],
which is positive on R for α ≥ 3. This shows (c) for n = 3.

⊓
⊔

Remark 2.20. Open question: In the notation of Example 2.19, is hα monotone of
⊓
⊔
order α + 1 for every α > 0 ?