getdoc45d6. 177KB Jun 04 2011 12:04:19 AM
Elect. Comm. in Probab. 15 (2010), 227–239
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
COMPOSITIONS OF MAPPINGS
OF INFINITELY DIVISIBLE DISTRIBUTIONS
WITH APPLICATIONS TO FINDING
THE LIMITS OF SOME NESTED SUBCLASSES
MAKOTO MAEJIMA
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi,
Kohoku-ku, Yokohama 223-8522, Japan
email: maejima@math.keio.ac.jp
YOHEI UEDA
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi,
Kohoku-ku, Yokohama 223-8522, Japan
email: ueda@2008.jukuin.keio.ac.jp
Submitted November 30, 2009, accepted in final form May 27, 2010
AMS 2000 Subject classification: 60E07
Keywords: infinitely divisible distribution on Rd , stochastic integral mapping, composition of mappings, limit of nested subclasses
Abstract
(µ)
Let {X t , t ≥ 0} be a Lévy process on Rd whose distribution at time 1 is µ, and let f be a
nonrandom measurable function on (0, a), 0 < a ≤ ∞. Then we can define a mapping Φ f (µ) by
Ra
(µ)
the law of 0 f (t)d X t , from D(Φ f ) which is the totality of µ ∈ I(Rd ) such that the stochastic
Ra
(µ)
integral 0 f (t)d X t is definable, into a class of infinitely divisible distributions. For m ∈ N, let
m
Φ f be the m times composition of Φ f itself. Maejima and Sato (2009) proved that the limits
T∞
m
m
known f ’s. Maejima and Nakahara (2009) introduced
m=1 Φ f (D(Φ f )) are the same for severalT
∞
)) for such general f ’s are investigated
more general f ’s. In this paper, the limits m=1 Φm
(D(Φm
f
f
by using the idea of compositions of suitable mappings of infinitely divisible distributions.
1
Introduction
Let P (Rd ) be the class of all probability distributions on Rd . Throughout this paper, L (X ) denotes
b(z), z ∈ Rd , denotes the characteristic function
the law of an Rd -valued random variable X and µ
d
d
of µ ∈ P (R ). Also I(R ) denotes the class of all infinitely divisible distributions on Rd . Cµ (z), z ∈
Rd , denotes the cumulant function of µ ∈ I(Rd ), that is, Cµ (z) is the unique continuous function
b(z) = e Cµ (z) and Cµ (0) = 0. For µ ∈ I(Rd ) and t > 0, we call the distribution with
satisfying µ
b(z) t = e t Cµ (z) the t-th convolution of µ and write µ t for it. We use the
characteristic function µ
227
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Electronic Communications in Probability
Lévy-Khintchine triplet (A, ν, γ) of µ ∈ I(Rd ) in the sense that
Cµ (z) = −2−1 〈z, Az〉 + i〈γ, z〉
Z
e i〈z,x〉 − 1 − i〈z, x〉(1 + |x|2 )−1 ν(d x),
+
Rd
z ∈ Rd ,
where | · | and 〈·, ·〉 are the Euclidean norm and inner product on Rd , respectively, A is a symmetric
nonnegative-definite d × d matrix,
γ ∈ Rd and ν is a measure (called the Lévy measure) on
R
d
R satisfying ν({0}) = 0 and Rd (|x|2 ∧ 1)ν(d x) < ∞. When we want to emphasize the LévyKhintchine triplet, we write µ = µ(A,ν,γ) .
We use stochastic integrals with respect to Lévy processes {X t , t ≥ 0} of nonrandom measurable
Rt
functions f : [0, ∞) → R, which are 0 f (s)d X s , t ∈ [0, ∞). As the definition of stochastic integrals, we adopt the method in Sato [25, 26]. It is known that if f is locally square integrable on
Rt
[0, ∞), then 0 f (s)d X s , t ∈ [0, ∞), is definable for any Lévy process {X t }. The improper stochasRt
R∞
tic integral 0 f (s)d X s is defined as the limit in probability of 0 f (s)d X s as t → ∞ whenever
Rt
the limit exists. In our definition, { 0 f (s)d X s , t ∈ [0, ∞)} is an additive process in law, which is
Rt
not always càdlàg in t. If we take its càdlàg modification, the convergence of 0 f (s)d X s above is
equivalent to the almost sure convergence of the modification as t → ∞.
(µ)
(µ)
Let {X t , t ≥ 0} stand for a Lévy process on Rd with L (X 1 ) = µ. Using this Lévy process, we
can define a mapping
Z
Φ f (µ) = L
a
f
(µ)
(t)d X t
,
0
µ ∈ D(Φ f ) ⊂ I(Rd ),
(1.1)
for a nonrandom measurable function f : [0, a) → R, where 0 < a ≤ ∞ and D(Φ f ) is the domain
Ra
(µ)
of a mapping Φ f that is the class of µ ∈ I(Rd ) for which 0 f (t)d X t is definable in the sense
Ra
above. Also, D0 (Φ f ) denotes the totality of µ ∈ I(Rd ) satisfying 0 |Cµ ( f (t)z)|d t < ∞ for all
z ∈ Rd . For a mapping Φ f , R(Φ f ) is its range that is {Φ f (µ) : µ ∈ D(Φ f )}. When we consider
the composition of two mappings Φ f and Φ g , denoted by Φ g ◦ Φ f , the domain of Φ g ◦ Φ f is
D(Φ g ◦ Φ f ) = {µ ∈ I(Rd ) : µ ∈ D(Φ f ) and Φ f (µ) ∈ D(Φ g )}. For m ∈ N, Φmf means the m times
composition of Φ f itself.
A set H ⊂ P (Rd ) is said to be closed under type equivalence if L (X ) ∈ H implies L (aX + c) ∈ H
for a > 0 and c ∈ Rd . H ⊂ I(Rd ) is called completely closed in the strong sense (abbreviated as
c.c.s.s.) if H is closed under type equivalence, convolution, weak convergence and t-th convolution
for any t > 0.
We list below several known mappings. In the following, Ilog (Rd ) denotes the totality of µ ∈ I(Rd )
R
satisfying Rd log+ |x|µ(d x) < ∞, where log+ |x| = (log |x|) ∨ 0.
(1) U -mapping (Alf and O’Connor [1], Jurek [8]): Let
Z
U (µ) = L
!
1
(µ)
tdXt
0
,
µ ∈ D(U ) = I(Rd ),
and let U(Rd ) be the Jurek class on Rd . Then U(Rd ) = U (I(Rd )).
Compositions of mappings of infinitely divisible distributions
229
(2) Φ-mapping (Wolfe [30], Jurek and Vervaat [15], Sato and Yamazato [29]): Let
Z ∞
(µ)
e−t d X t
Φ(µ) = L
µ ∈ D(Φ) = Ilog (Rd ),
,
0
d
and let L(R ) be the class of selfdecomposable distributions on Rd . Then L(Rd ) = Φ(Ilog (Rd )).
(3) Υ-mapping (Barndorff-Nielsen and Thorbjørnsen [6], Barndorff-Nielsen et al. [4]): Let
!
Z1
1 (µ)
, µ ∈ D(Υ) = I(Rd ),
(1.2)
Υ(µ) = L
log d X t
t
0
and let B(Rd ) be the Goldie–Steutel–Bondesson class on Rd . Then B(Rd ) = Υ(I(Rd )).
R∞
2
(4) G -mapping (Maejima and Sato [18]): Let t = h(s) = s e−u du, s > 0, and denote its inverse
function by s = h∗ (t). Let
!
Z pπ/2
(µ)
h∗ (t)d X t
G (µ) = L
µ ∈ D(G ) = I(Rd ),
,
0
and let G(Rd ) be the class of generalized type G distributions on Rd . Then G(Rd ) = G (I(Rd )).
R∞
(5) Ψ-mapping (Barndorff-Nielsen et al. [4]): Let t = e(s) = s u−1 e−u du, s > 0, and denote its
inverse function by s = e∗ (t). Let
Z ∞
(µ)
e∗ (t)d X t
Ψ(µ) = L
,
0
µ ∈ D(Ψ) = Ilog (Rd ),
d
and let T (R ) be the Thorin class on Rd . Then T (Rd ) = Ψ(Ilog (Rd )).
R∞
2
(6) M -mapping (Aoyama et al. [3]): Let t = m(s) = s u−1 e−u du, s > 0, and denote its inverse
function by s = m∗ (t). Let
Z ∞
(µ)
m∗ (t)d X t
M (µ) = L
,
0
µ ∈ D(M ) = Ilog (Rd ).
We call M (Rd ) := M (Ilog (Rd )) the class M and it was actually introduced in Aoyama et al. [3] in
the symmetric case.
Remark 1.1. Jurek [13] introduced the mapping
Z
K
(e)
(µ) = L
∞
0
(µ)
t d X 1−e−t
,
which is the same as Υ in (1.2) by the time change of the driving Lévy process. In the same way,
it holds that
Z ∞
G (µ) = L
(µ)
0
t d X pπ/2−h(t)
.
Using this type of time change, we might avoid taking inverse functions as integrands of stochastic
integral mappings. However, recently in Sato [27], Barndorff-Nielsen et al. [5] and other papers, they have used stochastic integral mappings whose integrands are some inverse functions
and driving Lévy processes have original time parameter. In this paper, we also use this type of
expressions.
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Electronic Communications in Probability
Here we also introduce mappings Φα , α < 2, (O’Connor [21, 22], Jurek [9, 10, 11], Jurek and
Schreiber [14], Sato [27], Maejima et al. [16]). Let
Z
1
u−α−1 du,
t = ϕα (s) =
s ≥ 0,
s
and let s = ϕα∗ (t) be its inverse function. Define
Z
(µ)
ϕα∗ (t)d X t
Φα (µ) = L
Then,
Z
!
ϕα (0)
.
0
!
−1/α
(µ)
dXt
(1 + αt)
, when α < 0,
L
0
Z ∞
(µ)
−t
Φα (µ) = L
e dXt
,
when α = 0,
0
Z ∞
(µ)
L
−1/α
dXt
(1 + αt)
,
when 0 < α < 2.
−1/α
0
Furthermore, we introduce mappings Ψα,β , α < 2, β > 0. Let
Z
∞
t = Gα,β (s) =
β
u−α−1 e−u du,
s
s ≥ 0,
∗
and let s = Gα,β
(t) be its inverse function. Define
Z
Ψα,β (µ) = L
where
¨
Gα,β (0) =
!
Gα,β (0)
(µ)
∗
Gα,β
(t)d X t
,
0
β −1 Γ(−αβ −1 ), when α < 0,
∞,
when α ≥ 0.
These mappings are introduced first by Sato [27] for β = 1 and later by Maejima and Nakahara
[17] for general β > 0. Due to Sato [27], Maejima and Nakahara [17], we have the domains
D(Φα ) and D(Ψα,β ) as follows. Let β > 0.
I(Rd ),
when α < 0,
d
I
(R
),
when
α = 0,
log
d
D(Φα ) = D(Ψα,β ) = Iα (R ), when 0 < α < 1,
∗
d
when α = 1,
I1 (R ),
0
Iα (Rd ), when 1 < α < 2,
Compositions of mappings of infinitely divisible distributions
231
where
¨
α
d
Iα (R ) = µ ∈ I(R ):
Rd
¨
Iα0 (Rd )
«
Z
d
xµ(d x) = 0 ,
= µ ∈ Iα (R ):
=
Rd
µ = µ(A,ν,γ) ∈
for α > 0,
«
Z
d
(
I1∗ (Rd )
|x| µ(d x) < ∞ ,
Z
I10 (Rd ): lim
T →∞
for α ≥ 1,
t
1
)
Z
T
−1
xν(d x) exists in R
dt
d
.
|x|>t
Since Φ0 = Φ, Φ−1 = U , Ψ−1,1 = Υ, Ψ−1,2 = G , Ψ0,1 = Ψ and Ψ0,2 = M , the mappings Φα and
Ψα,β are important. Also, Maejima and Nakahara [17] characterized the classes R(Ψα,β ), α <
1, β > 0 by conditions of radial components in the polar decomposition of Lévy measures.
Define nested subclasses L m (Rd ), m ∈ Z+ of L(Rd ) in the following way: µ ∈ L m (Rd ) if and only if
b(z) = µ
b(b−1 z)ρ
b b (z), where L0 (Rd ) :=
for each b > 1, there exists ρ b ∈ L m−1 (Rd ) such that µ
d
L(R ). Hereafter we denote the closure under weak convergence and convolution of a class
d
H ⊂ P (Rd ) by H.
of all α-stable distributions on Rd
S For α ∈ d(0, 2], let Sα (R ) be the dclass T
∞
d
and let S(R ) = α∈(0,2] Sα (R ). Then, the limit L∞ (R ) := m=0 L m (Rd ) is known to be equal
to S(Rd ). In Sato [24] or Rocha-Arteaga and Sato [23], this is proved via the following fact:
µ = µ(A,ν,γ) ∈ L∞ (Rd ) if and only if
Z
Z
ν(B) =
Z
Γ(dα)
(0,2)
λα (dξ)
∞
11B (rξ)r −α−1 d r,
0
S
B ∈ B(Rd \ {0}),
where Γ is a measure on (0, 2) satisfying
Z
(0,2)
1
α
+
1
2−α
Γ(dα) < ∞,
and λα is a probability measure on S := {ξ ∈ Rd : |ξ| = 1} for each α ∈ (0, 2), and λα (C) is
measurable in α ∈ (0, 2) for every C ∈ B(S). This Γ is uniquely determined by µ and this λα
is uniquely determined by µ up to α of Γ-measure 0. For the case in more general spaces, see
A
Jurek [7]. For a set A ∈ B ((0, 2)), let L∞
(Rd ) denote the class of µ ∈ L∞ (Rd ) with
T∞ Γ satisfying
m
d
Γ ((0, 2) \ A) = 0. It is also known that for m ∈ N, R(Φ ) = L m−1 (R ). Hence m=1 R(Φm ) =
L∞ (Rd ) = S(Rd ). In Maejima and Sato [18], nested subclasses R(U m ), R(Υm ), R(Ψm ) and
R(G m ), m ∈ N, were studied and the limits of these nested subclasses were proved to be equal to
S(Rd ), (see also Jurek [12]). Furthermore, Sato [28] proved that the mappings Ψα,1 , α ∈ (0, 2)
produce smaller classes than S(Rd ) as the limit of iteration. Maejima and Ueda [19] showed that
the mapping Φα has the same iterated limit as that of Ψα,1 for α ∈ (0, 2). Maejima and Ueda [20]
also constructed a mapping producing a larger class than S(Rd ), which is the closure of the class
of semi-stable distributions with a fixed span.
m
The purpose of this paper is to find the limit of the nested subclasses R(Ψα,β
), m ∈ N. For that,
we start with the composition of Ψα−β,β and Φα , which will be used for characterizing the nested
m
subclasses R(Ψα,β
), m ∈ N.
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Electronic Communications in Probability
2
Results
For β > 0, let
Z
Kβ (µ) = µ = L
!
β
(µ)
dXt
β
,
0
µ ∈ D(Kβ ) = I(Rd ).
The following lemma is trivial.
Lemma 2.1. For β > 0 and any mapping Φ f defined by (1.1) with a locally square-integrable
function f , we have
Kβ ◦ Φ f = Φ f ◦ Kβ .
Here
D(Kβ ◦ Φ f ) = D(Φ f ◦ Kβ ) = D(Φ f ).
The following result on composition will be a key in the proof of the main theorem, Theorem 2.4.
Theorem 2.2. Let α ∈ (−∞, 1) ∪ (1, 2) and β > 0. Then
Ψα,β = Kβ ◦ Φα ◦ Ψα−β,β = Kβ ◦ Ψα−β,β ◦ Φα ,
including the equality of the domains.
Remark 2.3. Theorem 2.2 with β = 1 is included in Theorem 3.1 of Sato [27]. Also, the case
α = 0 was already proved by Aoyama et al. [2].
Our main result of
is the following theorem on the limits of the nested subclasses
T∞this paper
m
m
) which is m=1 R(Ψα,β
).
R(Ψα,β
Theorem 2.4. Let β > 0. Then
d
for α ∈ (−∞, 0],
L∞ (R ),
m
(α,2)
d
R(Ψα,β ) = L∞ (R ),
for α ∈ (0, 1),
(α,2)
d
0
d
m=1
L∞ (R ) ∩ I1 (R ), for α ∈ (1, 2) \ {1 + nβ : n ∈ N}.
∞
\
Remark 2.5. Theorem 2.4 for the case −1 ≤ α < 0, β > 0 follows immediately from Theorem 3.4
of Maejima and Sato [18]. Maejima and Sato [18] also proved the case α = 0, β = 1. Furthermore,
the case β = 1, α ∈ (0, 2) was already proved by Sato [28]. The case α = 0 is found in Aoyama
et al. [2].
We also have the following.
Theorem 2.6. Let β > 0 and α ∈ (−∞, 2) \ {1 + nβ : n ∈ Z+ }. Then
∞
\
m=1
m
R(Ψα,β
) = R(Ψα,β ) ∩ S(Rd ).
(2.1)
Compositions of mappings of infinitely divisible distributions
3
233
Proofs
We first prove Theorem 2.2.
Proof of Theorem 2.2. For µ ∈ I(Rd ), we have
Z
Z
ϕα (0)
β
∗
(v)ϕα∗ (u)z d v
Cµ Gα−β,β
Gα−β,β (0)
du
0
0
Z
=β
Z
=
Z
∞
s−β−1 ds
Z
0
∞
0
∞
0
∞
C (tsz) t β−α−1 e−t β d t
µ
0
0
1
=β
=
∞
s−α−1 ds
Z
Z
Z
1
=β
C (uz) uβ−α−1 e−s−β uβ du
µ
0
C (uz) uβ−α−1 du
µ
C (uz) u−α−1 du
µ
Z
(3.1)
1
s
−β−1 −s−β uβ
e
ds
0
Z
∞
e−v d v
uβ
C (uz) u−α−1 e−uβ du =
µ
Z
∗
G
(t)z
d t.
Cµ
α,β
Gα,β (0)
0
0
Let α < 0. Then D(Ψα,β ) = D(Kβ ◦ Ψα−β,β ◦ Φα ) = D(Kβ ◦ Φα ◦ Ψα−β,β ) = I(Rd ) and Propositions
3.4 and 2.17 of Sato [26] yields the finiteness of (3.1). Then we can use Fubini’s theorem and
have
Z Gα,β (0)
Z Gα−β,β (0)
Z ϕα (0)
∗
Cµ Gα,β
(t)z d t = β
0
∗
Cµ Gα−β,β
(v)ϕα∗ (u)z d v
du
Z
0
0
Gα−β,β (0)
=β
Z
Cµ
dv
0
ϕα (0)
∗
Gα−β,β
(v)ϕα∗ (u)z
(3.2)
du,
0
by a similar calculation to (3.1). This yields that
Ψα,β (µ) = Kβ ◦ Φα ◦ Ψα−β,β (µ) = Kβ ◦ Ψα−β,β ◦ Φα (µ).
(3.3)
Let α ∈ [0, 1) ∪ (1, 2). Let µ ∈ D(Ψα,β ). Note that the domains D(Ψα,β ) and D(Φα ) are the same
and decreasing in α < 2 with respect to set inclusion due to Remark to Theorem 2.8 of Sato [27].
Then µ ∈ D(Ψα−β,β ) ∩ D(Φα ). We have that
Z
ϕα (0)
CΨα−β,β (µ) (ϕα∗ (u)z) du
β
0
and
Z
Gα−β,β (0)
∗
(G
(v)z)
CΦα (µ) α−β,β
dv
β
0
are not greater than (3.1). Take into account that D(Ψα,β ) = D0 (Ψα,β ) for α ∈ (−∞, 1) ∪ (1, 2)
and β > 0 due to Theorem 2.4 of Sato [27]. Then µ ∈ D0 (Ψα,β ), which yields the finiteness of
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Electronic Communications in Probability
(3.1). Therefore Ψα−β,β (µ) ∈ D(Kβ ◦ Φα ) and Φα (µ) ∈ D(Kβ ◦ Ψα−β,β ). Hence µ ∈ D(Kβ ◦ Φα ◦
Ψα−β,β ) ∩ D(Kβ ◦ Ψα−β,β ◦ Φα ).
If µ ∈ D(Kβ ◦ Ψα−β,β ◦ Φα ), then µ ∈ D(Φα ) = D(Ψα,β ).
e=
Let µ ∈ D(Kβ ◦ Φα ◦ Ψα−β,β ). Then Ψα−β,β (µ) ∈ D(Φα ). Let µ = µ(A,ν,γ) and Ψα−β,β (µ) = µ
e(Ae,eν ,eγ) . If α = 0, then
µ
Z
Z
∞>
|x|>1
log |x|e
ν (d x) =
Z
∞
t
β−1 −t β
e
log |t x|ν(d x),
dt
|t x|>1
0
R
which yields that |t x|>1 log |t x|ν(d x) < ∞ a.e. t > 0. Hence µ ∈ Ilog (Rd ) = D(Ψ0,β ). If α ∈
(0, 1) ∪ (1, 2), then
Z
Z∞
Z
∞>
which yields that
|x|>1
R
β
t β−α−1 e−t d t
|x|α νe(d x) =
|t x|>1
0
|x|α ν(d x) < ∞ a.e. t > 0. Hence µ ∈ Iα (Rd ) = D(Ψα,β ) for α ∈ (0, 1).
|t x|>1
e ∈ D(Φα ) = Iα0 (Rd ). It follows that
Let α ∈ (1, 2). Then µ
Z
e=−
γ
and
x|x|2
1 + |x|
Rd
Z
Z
νe(d x) = −
2
¨
∞
e = lim
γ
ǫ↓0
|t x|α ν(d x),
t
β−α −t β
e
dt γ +
lim
t
β−α −t β
e
e
x
Z
1
1 + |t x|2
x|x|2
γ+
Rd
ǫ
t x|t x|2
dt
Rd
∞
Z
β−α−1 −t β
0
Rd
Z
ǫ↓0
t
Z
ǫ
Hence
∞
1 + |x|2
−
1 + |t x|2
1
1 + |x|2
ν(d x)
«
ν(d x) .
ν(d x) d t = 0,
R x|x|2
which yields that γ + Rd 1+|x|2 ν(d x) = 0. Therefore µ ∈ Iα0 (Rd ) = D(Ψα,β ).
Thus we conclude that D(Ψα,β ) = D(Kβ ◦Ψα−β,β ◦Φα ) = D(Kβ ◦Φα ◦Ψα−β,β ) for α ∈ [0, 1)∪(1, 2).
If µ ∈ D(Ψα,β ) = D0 (Ψα,β ), then (3.1) is finite and we have (3.2) and (3.3).
Let
(
d
Cα (R ) :=
)
Z
d
µ = µ(A,ν,γ) ∈ I(R ): lim r
r→∞
Cα0 (Rd ) := Cα (Rd ) ∩ I10 (Rd ).
α
ν(d x) = 0 ,
|x|>r
We need the following lemma in the proof below.
Lemma 3.1 (Corollary 4.2 of Maejima and Ueda [19]). Let H ⊂ I(Rd ) be c.c.s.s.
(i) If α ≤ 0 and H ⊃ S(Rd ), then
∞
\
m=1
Φαm (H ∩ D(Φαm )) = S(Rd ).
Compositions of mappings of infinitely divisible distributions
(ii) If 0 < α < 1 and H ⊃
S
∞
\
m=1
), then
β∈[α,2]
S
∞
\
d
[
Sβ (Rd ) ∩ Cα (Rd ).
Φαm H ∩ D(Φαm ) =
m=1
(iii) If 1 < α < 2 and H ⊃
β∈[α,2] Sβ (R
235
β∈[α,2] Sβ (R
d
), then
[
Φαm H ∩ D(Φαm ) =
Sβ (Rd ) ∩ Cα0 (Rd ).
β∈[α,2]
S
Here S(Rd ) and β∈[α,2] Sβ (Rd ) are c.c.s.s., (see e.g. Proposition 3.12 (i) and Theorem 3.20 of
Maejima and Ueda [19]).
To prove Theorem 2.4, it is sufficient to show the following, due to Theorem 4.6 of Maejima and
Ueda [19] that is Theorem 2.4 with the replacement of Ψα,β by Φα .
Theorem 3.2. Let β > 0 and α ∈ (−∞, 2) \ {1 + nβ : n ∈ Z+ }. Then we have
∞
\
m
R(Ψα,β
)=
∞
\
R(Φαm ).
(3.4)
m=1
m=1
Proof. Lemma 2.1 and Theorem 2.2 yield that for α ∈ (−∞, 1) ∪ (1, 2), β > 0 and m ∈ N,
m
m
R(Ψα,β
) = R(Φαm ◦ Ψα−β,β
◦ Kβm )
m
= Φαm R(Ψα−β,β
◦ Kβm ) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
R(Kβm ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
R(Kβ m ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
I(Rd ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
= Φαm R(Ψα−β,β
) ∩ D(Φαm ) .
(3.5)
Fix any β > 0.
We first show
∞
\
m=1
m
R(Ψα,β
)=
∞
\
m=1
R(Φαm ) for all α < (nβ) ∧ 1,
(3.6)
for each n ∈ N byTinduction. Let n = 1. For α < β ∧ 1, Proposition 3.2 of Maejima and Sato [18]
∞
k
entails that H := k=1 R(Ψα−β,β
) is c.c.s.s. Also, Lemma 3.7 of Maejima and Sato [18] yields that
H ⊃ S(Rd ). It follows from (3.5) that for m ∈ N,
m
) ⊂ R(Φαm ).
Φαm H ∩ D(Φαm ) ⊂ R(Ψα,β
Thus Lemma 3.1 yields (3.6) with n = 1. Assume that (3.6) is true for n − 1 in place of n with
236
Electronic Communications in Probability
n ≥ 2. Then for α < (nβ) ∧ 1, it follows that α − β < ((n − 1)β) ∧ 1. Therefore
∞
\
k
R(Ψα−β,β
)=
k=1
(
=
∞
\
k
R(Φα−β
)
k=1
S(Rd ),
S
γ∈[α−β,2] Sγ
(Rd ) ∩ C
α−β (R
d
),
when α − β ≤ 0
when 0 < α − β < 1,
by the assumption of induction and Lemma 3.1. When α − β ≤ 0, it follows from (3.5) that for
m ∈ N,
m
Φαm S(Rd ) ∩ D(Φαm ) ⊂ R(Ψα,β
) ⊂ R(Φαm ),
which yields (3.6) by Lemma 3.1. When 0 < α − β < 1, it follows from (3.5) that for m ∈ N,
[
m
) ⊂ R(Φαm ),
(3.7)
Sγ (Rd ) ∩ D(Φαm ) ⊂ R(Ψα,β
Φαm
γ∈[α−β,2]
since D(Φαm ) ⊂ D(Φα ) ⊂ Iα (Rd ) ⊂ Iα−β (Rd ) ⊂ Cα−β (Rd ). Thus Lemma 3.1 yields (3.6). Then
(3.6) is true for all n ∈ N, namely, (3.4) holds for all α < 1.
We next show
∞
\
∞
\
m
R(Ψα,β
)=
m=1
m=1
R(Φαm ) for all α ∈ 1 + (n − 1)β, 1 + nβ ∩ (−∞, 2),
(3.8)
for each n ∈ Z+ by induction. If n = 0, then α < 1 and we have just shown the case.
Assume that
(3.8) holds for n − 1 in place of n with n ≥ 1. Then
for
α
∈
1
+
(n
−
1)β,
1
+
nβ
∩ (−∞, 2), it
follows that α − β ∈ 1 + (n − 2)β, 1 + (n − 1)β ∩ (−∞, 2). Then the assumption of induction
and Lemma 3.1 yields that
∞
\
k
R(Ψα−β,β
)=
k=1
=
∞
\
k
R(Φα−β
)
k=1
S(Rd ),
S
Sγ∈[α−β,2]
Sγ
(Rd ) ∩ C
α−β (R
d
),
d
0
d
γ∈[α−β,2] Sγ (R ) ∩ Cα−β (R ),
when α − β ≤ 0
when 0 < α − β < 1,
when 1 < α − β < 2.
When α − β < 1, (3.8) holds by the same argument as above. When 1 < α − β < 2, the same
0
0
(Rd ).
(Rd ) ⊂ Cα−β
inclusion as (3.7) follows from (3.5), since D(Φαm ) ⊂ D(Φα ) ⊂ Iα0 (Rd ) ⊂ Iα−β
Therefore Lemma 3.1 yields (3.8). Then (3.8) is true for all n ∈ Z+ . Thus (3.4) holds for all
α ∈ (1, 2) \ {1 + nβ : n ∈ N}.
We finally prove Theorem 2.6.
Proof of Theorem 2.6. (3.5) with m = 1 yields that R(Ψα,β ) ⊂ R(Φα ). Also we have
∞
\
m=1
R(Φαm ) = R(Φα ) ∩ S(Rd ),
Compositions of mappings of infinitely divisible distributions
237
which is Theorem 5.2 of Maejima and Ueda [19]. Then we have
R(Ψα,β ) ∩ S(Rd ) ⊂ R(Φα ) ∩ S(Rd ) =
∞
\
R(Φαm ) =
m=1
∞
\
m
R(Ψα,β
),
m=1
where the last equality follows from Theorem 3.2. Furthermore, we have that
∞
\
m=1
and that
∞
\
m=1
Thus
T∞
m
m=1 R(Ψα,β )
m
R(Ψα,β
)=
m
R(Ψα,β
) ⊂ R(Ψα,β ),
∞
\
m=1
R(Φαm ) = R(Φα ) ∩ S(Rd ) ⊂ S(Rd ).
⊂ R(Ψα,β ) ∩ S(Rd ). Therefore (2.1) holds.
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239
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
COMPOSITIONS OF MAPPINGS
OF INFINITELY DIVISIBLE DISTRIBUTIONS
WITH APPLICATIONS TO FINDING
THE LIMITS OF SOME NESTED SUBCLASSES
MAKOTO MAEJIMA
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi,
Kohoku-ku, Yokohama 223-8522, Japan
email: maejima@math.keio.ac.jp
YOHEI UEDA
Department of Mathematics, Faculty of Science and Technology, Keio University, 3-14-1, Hiyoshi,
Kohoku-ku, Yokohama 223-8522, Japan
email: ueda@2008.jukuin.keio.ac.jp
Submitted November 30, 2009, accepted in final form May 27, 2010
AMS 2000 Subject classification: 60E07
Keywords: infinitely divisible distribution on Rd , stochastic integral mapping, composition of mappings, limit of nested subclasses
Abstract
(µ)
Let {X t , t ≥ 0} be a Lévy process on Rd whose distribution at time 1 is µ, and let f be a
nonrandom measurable function on (0, a), 0 < a ≤ ∞. Then we can define a mapping Φ f (µ) by
Ra
(µ)
the law of 0 f (t)d X t , from D(Φ f ) which is the totality of µ ∈ I(Rd ) such that the stochastic
Ra
(µ)
integral 0 f (t)d X t is definable, into a class of infinitely divisible distributions. For m ∈ N, let
m
Φ f be the m times composition of Φ f itself. Maejima and Sato (2009) proved that the limits
T∞
m
m
known f ’s. Maejima and Nakahara (2009) introduced
m=1 Φ f (D(Φ f )) are the same for severalT
∞
)) for such general f ’s are investigated
more general f ’s. In this paper, the limits m=1 Φm
(D(Φm
f
f
by using the idea of compositions of suitable mappings of infinitely divisible distributions.
1
Introduction
Let P (Rd ) be the class of all probability distributions on Rd . Throughout this paper, L (X ) denotes
b(z), z ∈ Rd , denotes the characteristic function
the law of an Rd -valued random variable X and µ
d
d
of µ ∈ P (R ). Also I(R ) denotes the class of all infinitely divisible distributions on Rd . Cµ (z), z ∈
Rd , denotes the cumulant function of µ ∈ I(Rd ), that is, Cµ (z) is the unique continuous function
b(z) = e Cµ (z) and Cµ (0) = 0. For µ ∈ I(Rd ) and t > 0, we call the distribution with
satisfying µ
b(z) t = e t Cµ (z) the t-th convolution of µ and write µ t for it. We use the
characteristic function µ
227
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Electronic Communications in Probability
Lévy-Khintchine triplet (A, ν, γ) of µ ∈ I(Rd ) in the sense that
Cµ (z) = −2−1 〈z, Az〉 + i〈γ, z〉
Z
e i〈z,x〉 − 1 − i〈z, x〉(1 + |x|2 )−1 ν(d x),
+
Rd
z ∈ Rd ,
where | · | and 〈·, ·〉 are the Euclidean norm and inner product on Rd , respectively, A is a symmetric
nonnegative-definite d × d matrix,
γ ∈ Rd and ν is a measure (called the Lévy measure) on
R
d
R satisfying ν({0}) = 0 and Rd (|x|2 ∧ 1)ν(d x) < ∞. When we want to emphasize the LévyKhintchine triplet, we write µ = µ(A,ν,γ) .
We use stochastic integrals with respect to Lévy processes {X t , t ≥ 0} of nonrandom measurable
Rt
functions f : [0, ∞) → R, which are 0 f (s)d X s , t ∈ [0, ∞). As the definition of stochastic integrals, we adopt the method in Sato [25, 26]. It is known that if f is locally square integrable on
Rt
[0, ∞), then 0 f (s)d X s , t ∈ [0, ∞), is definable for any Lévy process {X t }. The improper stochasRt
R∞
tic integral 0 f (s)d X s is defined as the limit in probability of 0 f (s)d X s as t → ∞ whenever
Rt
the limit exists. In our definition, { 0 f (s)d X s , t ∈ [0, ∞)} is an additive process in law, which is
Rt
not always càdlàg in t. If we take its càdlàg modification, the convergence of 0 f (s)d X s above is
equivalent to the almost sure convergence of the modification as t → ∞.
(µ)
(µ)
Let {X t , t ≥ 0} stand for a Lévy process on Rd with L (X 1 ) = µ. Using this Lévy process, we
can define a mapping
Z
Φ f (µ) = L
a
f
(µ)
(t)d X t
,
0
µ ∈ D(Φ f ) ⊂ I(Rd ),
(1.1)
for a nonrandom measurable function f : [0, a) → R, where 0 < a ≤ ∞ and D(Φ f ) is the domain
Ra
(µ)
of a mapping Φ f that is the class of µ ∈ I(Rd ) for which 0 f (t)d X t is definable in the sense
Ra
above. Also, D0 (Φ f ) denotes the totality of µ ∈ I(Rd ) satisfying 0 |Cµ ( f (t)z)|d t < ∞ for all
z ∈ Rd . For a mapping Φ f , R(Φ f ) is its range that is {Φ f (µ) : µ ∈ D(Φ f )}. When we consider
the composition of two mappings Φ f and Φ g , denoted by Φ g ◦ Φ f , the domain of Φ g ◦ Φ f is
D(Φ g ◦ Φ f ) = {µ ∈ I(Rd ) : µ ∈ D(Φ f ) and Φ f (µ) ∈ D(Φ g )}. For m ∈ N, Φmf means the m times
composition of Φ f itself.
A set H ⊂ P (Rd ) is said to be closed under type equivalence if L (X ) ∈ H implies L (aX + c) ∈ H
for a > 0 and c ∈ Rd . H ⊂ I(Rd ) is called completely closed in the strong sense (abbreviated as
c.c.s.s.) if H is closed under type equivalence, convolution, weak convergence and t-th convolution
for any t > 0.
We list below several known mappings. In the following, Ilog (Rd ) denotes the totality of µ ∈ I(Rd )
R
satisfying Rd log+ |x|µ(d x) < ∞, where log+ |x| = (log |x|) ∨ 0.
(1) U -mapping (Alf and O’Connor [1], Jurek [8]): Let
Z
U (µ) = L
!
1
(µ)
tdXt
0
,
µ ∈ D(U ) = I(Rd ),
and let U(Rd ) be the Jurek class on Rd . Then U(Rd ) = U (I(Rd )).
Compositions of mappings of infinitely divisible distributions
229
(2) Φ-mapping (Wolfe [30], Jurek and Vervaat [15], Sato and Yamazato [29]): Let
Z ∞
(µ)
e−t d X t
Φ(µ) = L
µ ∈ D(Φ) = Ilog (Rd ),
,
0
d
and let L(R ) be the class of selfdecomposable distributions on Rd . Then L(Rd ) = Φ(Ilog (Rd )).
(3) Υ-mapping (Barndorff-Nielsen and Thorbjørnsen [6], Barndorff-Nielsen et al. [4]): Let
!
Z1
1 (µ)
, µ ∈ D(Υ) = I(Rd ),
(1.2)
Υ(µ) = L
log d X t
t
0
and let B(Rd ) be the Goldie–Steutel–Bondesson class on Rd . Then B(Rd ) = Υ(I(Rd )).
R∞
2
(4) G -mapping (Maejima and Sato [18]): Let t = h(s) = s e−u du, s > 0, and denote its inverse
function by s = h∗ (t). Let
!
Z pπ/2
(µ)
h∗ (t)d X t
G (µ) = L
µ ∈ D(G ) = I(Rd ),
,
0
and let G(Rd ) be the class of generalized type G distributions on Rd . Then G(Rd ) = G (I(Rd )).
R∞
(5) Ψ-mapping (Barndorff-Nielsen et al. [4]): Let t = e(s) = s u−1 e−u du, s > 0, and denote its
inverse function by s = e∗ (t). Let
Z ∞
(µ)
e∗ (t)d X t
Ψ(µ) = L
,
0
µ ∈ D(Ψ) = Ilog (Rd ),
d
and let T (R ) be the Thorin class on Rd . Then T (Rd ) = Ψ(Ilog (Rd )).
R∞
2
(6) M -mapping (Aoyama et al. [3]): Let t = m(s) = s u−1 e−u du, s > 0, and denote its inverse
function by s = m∗ (t). Let
Z ∞
(µ)
m∗ (t)d X t
M (µ) = L
,
0
µ ∈ D(M ) = Ilog (Rd ).
We call M (Rd ) := M (Ilog (Rd )) the class M and it was actually introduced in Aoyama et al. [3] in
the symmetric case.
Remark 1.1. Jurek [13] introduced the mapping
Z
K
(e)
(µ) = L
∞
0
(µ)
t d X 1−e−t
,
which is the same as Υ in (1.2) by the time change of the driving Lévy process. In the same way,
it holds that
Z ∞
G (µ) = L
(µ)
0
t d X pπ/2−h(t)
.
Using this type of time change, we might avoid taking inverse functions as integrands of stochastic
integral mappings. However, recently in Sato [27], Barndorff-Nielsen et al. [5] and other papers, they have used stochastic integral mappings whose integrands are some inverse functions
and driving Lévy processes have original time parameter. In this paper, we also use this type of
expressions.
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Electronic Communications in Probability
Here we also introduce mappings Φα , α < 2, (O’Connor [21, 22], Jurek [9, 10, 11], Jurek and
Schreiber [14], Sato [27], Maejima et al. [16]). Let
Z
1
u−α−1 du,
t = ϕα (s) =
s ≥ 0,
s
and let s = ϕα∗ (t) be its inverse function. Define
Z
(µ)
ϕα∗ (t)d X t
Φα (µ) = L
Then,
Z
!
ϕα (0)
.
0
!
−1/α
(µ)
dXt
(1 + αt)
, when α < 0,
L
0
Z ∞
(µ)
−t
Φα (µ) = L
e dXt
,
when α = 0,
0
Z ∞
(µ)
L
−1/α
dXt
(1 + αt)
,
when 0 < α < 2.
−1/α
0
Furthermore, we introduce mappings Ψα,β , α < 2, β > 0. Let
Z
∞
t = Gα,β (s) =
β
u−α−1 e−u du,
s
s ≥ 0,
∗
and let s = Gα,β
(t) be its inverse function. Define
Z
Ψα,β (µ) = L
where
¨
Gα,β (0) =
!
Gα,β (0)
(µ)
∗
Gα,β
(t)d X t
,
0
β −1 Γ(−αβ −1 ), when α < 0,
∞,
when α ≥ 0.
These mappings are introduced first by Sato [27] for β = 1 and later by Maejima and Nakahara
[17] for general β > 0. Due to Sato [27], Maejima and Nakahara [17], we have the domains
D(Φα ) and D(Ψα,β ) as follows. Let β > 0.
I(Rd ),
when α < 0,
d
I
(R
),
when
α = 0,
log
d
D(Φα ) = D(Ψα,β ) = Iα (R ), when 0 < α < 1,
∗
d
when α = 1,
I1 (R ),
0
Iα (Rd ), when 1 < α < 2,
Compositions of mappings of infinitely divisible distributions
231
where
¨
α
d
Iα (R ) = µ ∈ I(R ):
Rd
¨
Iα0 (Rd )
«
Z
d
xµ(d x) = 0 ,
= µ ∈ Iα (R ):
=
Rd
µ = µ(A,ν,γ) ∈
for α > 0,
«
Z
d
(
I1∗ (Rd )
|x| µ(d x) < ∞ ,
Z
I10 (Rd ): lim
T →∞
for α ≥ 1,
t
1
)
Z
T
−1
xν(d x) exists in R
dt
d
.
|x|>t
Since Φ0 = Φ, Φ−1 = U , Ψ−1,1 = Υ, Ψ−1,2 = G , Ψ0,1 = Ψ and Ψ0,2 = M , the mappings Φα and
Ψα,β are important. Also, Maejima and Nakahara [17] characterized the classes R(Ψα,β ), α <
1, β > 0 by conditions of radial components in the polar decomposition of Lévy measures.
Define nested subclasses L m (Rd ), m ∈ Z+ of L(Rd ) in the following way: µ ∈ L m (Rd ) if and only if
b(z) = µ
b(b−1 z)ρ
b b (z), where L0 (Rd ) :=
for each b > 1, there exists ρ b ∈ L m−1 (Rd ) such that µ
d
L(R ). Hereafter we denote the closure under weak convergence and convolution of a class
d
H ⊂ P (Rd ) by H.
of all α-stable distributions on Rd
S For α ∈ d(0, 2], let Sα (R ) be the dclass T
∞
d
and let S(R ) = α∈(0,2] Sα (R ). Then, the limit L∞ (R ) := m=0 L m (Rd ) is known to be equal
to S(Rd ). In Sato [24] or Rocha-Arteaga and Sato [23], this is proved via the following fact:
µ = µ(A,ν,γ) ∈ L∞ (Rd ) if and only if
Z
Z
ν(B) =
Z
Γ(dα)
(0,2)
λα (dξ)
∞
11B (rξ)r −α−1 d r,
0
S
B ∈ B(Rd \ {0}),
where Γ is a measure on (0, 2) satisfying
Z
(0,2)
1
α
+
1
2−α
Γ(dα) < ∞,
and λα is a probability measure on S := {ξ ∈ Rd : |ξ| = 1} for each α ∈ (0, 2), and λα (C) is
measurable in α ∈ (0, 2) for every C ∈ B(S). This Γ is uniquely determined by µ and this λα
is uniquely determined by µ up to α of Γ-measure 0. For the case in more general spaces, see
A
Jurek [7]. For a set A ∈ B ((0, 2)), let L∞
(Rd ) denote the class of µ ∈ L∞ (Rd ) with
T∞ Γ satisfying
m
d
Γ ((0, 2) \ A) = 0. It is also known that for m ∈ N, R(Φ ) = L m−1 (R ). Hence m=1 R(Φm ) =
L∞ (Rd ) = S(Rd ). In Maejima and Sato [18], nested subclasses R(U m ), R(Υm ), R(Ψm ) and
R(G m ), m ∈ N, were studied and the limits of these nested subclasses were proved to be equal to
S(Rd ), (see also Jurek [12]). Furthermore, Sato [28] proved that the mappings Ψα,1 , α ∈ (0, 2)
produce smaller classes than S(Rd ) as the limit of iteration. Maejima and Ueda [19] showed that
the mapping Φα has the same iterated limit as that of Ψα,1 for α ∈ (0, 2). Maejima and Ueda [20]
also constructed a mapping producing a larger class than S(Rd ), which is the closure of the class
of semi-stable distributions with a fixed span.
m
The purpose of this paper is to find the limit of the nested subclasses R(Ψα,β
), m ∈ N. For that,
we start with the composition of Ψα−β,β and Φα , which will be used for characterizing the nested
m
subclasses R(Ψα,β
), m ∈ N.
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Electronic Communications in Probability
2
Results
For β > 0, let
Z
Kβ (µ) = µ = L
!
β
(µ)
dXt
β
,
0
µ ∈ D(Kβ ) = I(Rd ).
The following lemma is trivial.
Lemma 2.1. For β > 0 and any mapping Φ f defined by (1.1) with a locally square-integrable
function f , we have
Kβ ◦ Φ f = Φ f ◦ Kβ .
Here
D(Kβ ◦ Φ f ) = D(Φ f ◦ Kβ ) = D(Φ f ).
The following result on composition will be a key in the proof of the main theorem, Theorem 2.4.
Theorem 2.2. Let α ∈ (−∞, 1) ∪ (1, 2) and β > 0. Then
Ψα,β = Kβ ◦ Φα ◦ Ψα−β,β = Kβ ◦ Ψα−β,β ◦ Φα ,
including the equality of the domains.
Remark 2.3. Theorem 2.2 with β = 1 is included in Theorem 3.1 of Sato [27]. Also, the case
α = 0 was already proved by Aoyama et al. [2].
Our main result of
is the following theorem on the limits of the nested subclasses
T∞this paper
m
m
) which is m=1 R(Ψα,β
).
R(Ψα,β
Theorem 2.4. Let β > 0. Then
d
for α ∈ (−∞, 0],
L∞ (R ),
m
(α,2)
d
R(Ψα,β ) = L∞ (R ),
for α ∈ (0, 1),
(α,2)
d
0
d
m=1
L∞ (R ) ∩ I1 (R ), for α ∈ (1, 2) \ {1 + nβ : n ∈ N}.
∞
\
Remark 2.5. Theorem 2.4 for the case −1 ≤ α < 0, β > 0 follows immediately from Theorem 3.4
of Maejima and Sato [18]. Maejima and Sato [18] also proved the case α = 0, β = 1. Furthermore,
the case β = 1, α ∈ (0, 2) was already proved by Sato [28]. The case α = 0 is found in Aoyama
et al. [2].
We also have the following.
Theorem 2.6. Let β > 0 and α ∈ (−∞, 2) \ {1 + nβ : n ∈ Z+ }. Then
∞
\
m=1
m
R(Ψα,β
) = R(Ψα,β ) ∩ S(Rd ).
(2.1)
Compositions of mappings of infinitely divisible distributions
3
233
Proofs
We first prove Theorem 2.2.
Proof of Theorem 2.2. For µ ∈ I(Rd ), we have
Z
Z
ϕα (0)
β
∗
(v)ϕα∗ (u)z d v
Cµ Gα−β,β
Gα−β,β (0)
du
0
0
Z
=β
Z
=
Z
∞
s−β−1 ds
Z
0
∞
0
∞
0
∞
C (tsz) t β−α−1 e−t β d t
µ
0
0
1
=β
=
∞
s−α−1 ds
Z
Z
Z
1
=β
C (uz) uβ−α−1 e−s−β uβ du
µ
0
C (uz) uβ−α−1 du
µ
C (uz) u−α−1 du
µ
Z
(3.1)
1
s
−β−1 −s−β uβ
e
ds
0
Z
∞
e−v d v
uβ
C (uz) u−α−1 e−uβ du =
µ
Z
∗
G
(t)z
d t.
Cµ
α,β
Gα,β (0)
0
0
Let α < 0. Then D(Ψα,β ) = D(Kβ ◦ Ψα−β,β ◦ Φα ) = D(Kβ ◦ Φα ◦ Ψα−β,β ) = I(Rd ) and Propositions
3.4 and 2.17 of Sato [26] yields the finiteness of (3.1). Then we can use Fubini’s theorem and
have
Z Gα,β (0)
Z Gα−β,β (0)
Z ϕα (0)
∗
Cµ Gα,β
(t)z d t = β
0
∗
Cµ Gα−β,β
(v)ϕα∗ (u)z d v
du
Z
0
0
Gα−β,β (0)
=β
Z
Cµ
dv
0
ϕα (0)
∗
Gα−β,β
(v)ϕα∗ (u)z
(3.2)
du,
0
by a similar calculation to (3.1). This yields that
Ψα,β (µ) = Kβ ◦ Φα ◦ Ψα−β,β (µ) = Kβ ◦ Ψα−β,β ◦ Φα (µ).
(3.3)
Let α ∈ [0, 1) ∪ (1, 2). Let µ ∈ D(Ψα,β ). Note that the domains D(Ψα,β ) and D(Φα ) are the same
and decreasing in α < 2 with respect to set inclusion due to Remark to Theorem 2.8 of Sato [27].
Then µ ∈ D(Ψα−β,β ) ∩ D(Φα ). We have that
Z
ϕα (0)
CΨα−β,β (µ) (ϕα∗ (u)z) du
β
0
and
Z
Gα−β,β (0)
∗
(G
(v)z)
CΦα (µ) α−β,β
dv
β
0
are not greater than (3.1). Take into account that D(Ψα,β ) = D0 (Ψα,β ) for α ∈ (−∞, 1) ∪ (1, 2)
and β > 0 due to Theorem 2.4 of Sato [27]. Then µ ∈ D0 (Ψα,β ), which yields the finiteness of
234
Electronic Communications in Probability
(3.1). Therefore Ψα−β,β (µ) ∈ D(Kβ ◦ Φα ) and Φα (µ) ∈ D(Kβ ◦ Ψα−β,β ). Hence µ ∈ D(Kβ ◦ Φα ◦
Ψα−β,β ) ∩ D(Kβ ◦ Ψα−β,β ◦ Φα ).
If µ ∈ D(Kβ ◦ Ψα−β,β ◦ Φα ), then µ ∈ D(Φα ) = D(Ψα,β ).
e=
Let µ ∈ D(Kβ ◦ Φα ◦ Ψα−β,β ). Then Ψα−β,β (µ) ∈ D(Φα ). Let µ = µ(A,ν,γ) and Ψα−β,β (µ) = µ
e(Ae,eν ,eγ) . If α = 0, then
µ
Z
Z
∞>
|x|>1
log |x|e
ν (d x) =
Z
∞
t
β−1 −t β
e
log |t x|ν(d x),
dt
|t x|>1
0
R
which yields that |t x|>1 log |t x|ν(d x) < ∞ a.e. t > 0. Hence µ ∈ Ilog (Rd ) = D(Ψ0,β ). If α ∈
(0, 1) ∪ (1, 2), then
Z
Z∞
Z
∞>
which yields that
|x|>1
R
β
t β−α−1 e−t d t
|x|α νe(d x) =
|t x|>1
0
|x|α ν(d x) < ∞ a.e. t > 0. Hence µ ∈ Iα (Rd ) = D(Ψα,β ) for α ∈ (0, 1).
|t x|>1
e ∈ D(Φα ) = Iα0 (Rd ). It follows that
Let α ∈ (1, 2). Then µ
Z
e=−
γ
and
x|x|2
1 + |x|
Rd
Z
Z
νe(d x) = −
2
¨
∞
e = lim
γ
ǫ↓0
|t x|α ν(d x),
t
β−α −t β
e
dt γ +
lim
t
β−α −t β
e
e
x
Z
1
1 + |t x|2
x|x|2
γ+
Rd
ǫ
t x|t x|2
dt
Rd
∞
Z
β−α−1 −t β
0
Rd
Z
ǫ↓0
t
Z
ǫ
Hence
∞
1 + |x|2
−
1 + |t x|2
1
1 + |x|2
ν(d x)
«
ν(d x) .
ν(d x) d t = 0,
R x|x|2
which yields that γ + Rd 1+|x|2 ν(d x) = 0. Therefore µ ∈ Iα0 (Rd ) = D(Ψα,β ).
Thus we conclude that D(Ψα,β ) = D(Kβ ◦Ψα−β,β ◦Φα ) = D(Kβ ◦Φα ◦Ψα−β,β ) for α ∈ [0, 1)∪(1, 2).
If µ ∈ D(Ψα,β ) = D0 (Ψα,β ), then (3.1) is finite and we have (3.2) and (3.3).
Let
(
d
Cα (R ) :=
)
Z
d
µ = µ(A,ν,γ) ∈ I(R ): lim r
r→∞
Cα0 (Rd ) := Cα (Rd ) ∩ I10 (Rd ).
α
ν(d x) = 0 ,
|x|>r
We need the following lemma in the proof below.
Lemma 3.1 (Corollary 4.2 of Maejima and Ueda [19]). Let H ⊂ I(Rd ) be c.c.s.s.
(i) If α ≤ 0 and H ⊃ S(Rd ), then
∞
\
m=1
Φαm (H ∩ D(Φαm )) = S(Rd ).
Compositions of mappings of infinitely divisible distributions
(ii) If 0 < α < 1 and H ⊃
S
∞
\
m=1
), then
β∈[α,2]
S
∞
\
d
[
Sβ (Rd ) ∩ Cα (Rd ).
Φαm H ∩ D(Φαm ) =
m=1
(iii) If 1 < α < 2 and H ⊃
β∈[α,2] Sβ (R
235
β∈[α,2] Sβ (R
d
), then
[
Φαm H ∩ D(Φαm ) =
Sβ (Rd ) ∩ Cα0 (Rd ).
β∈[α,2]
S
Here S(Rd ) and β∈[α,2] Sβ (Rd ) are c.c.s.s., (see e.g. Proposition 3.12 (i) and Theorem 3.20 of
Maejima and Ueda [19]).
To prove Theorem 2.4, it is sufficient to show the following, due to Theorem 4.6 of Maejima and
Ueda [19] that is Theorem 2.4 with the replacement of Ψα,β by Φα .
Theorem 3.2. Let β > 0 and α ∈ (−∞, 2) \ {1 + nβ : n ∈ Z+ }. Then we have
∞
\
m
R(Ψα,β
)=
∞
\
R(Φαm ).
(3.4)
m=1
m=1
Proof. Lemma 2.1 and Theorem 2.2 yield that for α ∈ (−∞, 1) ∪ (1, 2), β > 0 and m ∈ N,
m
m
R(Ψα,β
) = R(Φαm ◦ Ψα−β,β
◦ Kβm )
m
= Φαm R(Ψα−β,β
◦ Kβm ) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
R(Kβm ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
R(Kβ m ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
m
= Φαm Ψα−β,β
I(Rd ) ∩ D(Ψα−β,β
) ∩ D(Φαm )
m
= Φαm R(Ψα−β,β
) ∩ D(Φαm ) .
(3.5)
Fix any β > 0.
We first show
∞
\
m=1
m
R(Ψα,β
)=
∞
\
m=1
R(Φαm ) for all α < (nβ) ∧ 1,
(3.6)
for each n ∈ N byTinduction. Let n = 1. For α < β ∧ 1, Proposition 3.2 of Maejima and Sato [18]
∞
k
entails that H := k=1 R(Ψα−β,β
) is c.c.s.s. Also, Lemma 3.7 of Maejima and Sato [18] yields that
H ⊃ S(Rd ). It follows from (3.5) that for m ∈ N,
m
) ⊂ R(Φαm ).
Φαm H ∩ D(Φαm ) ⊂ R(Ψα,β
Thus Lemma 3.1 yields (3.6) with n = 1. Assume that (3.6) is true for n − 1 in place of n with
236
Electronic Communications in Probability
n ≥ 2. Then for α < (nβ) ∧ 1, it follows that α − β < ((n − 1)β) ∧ 1. Therefore
∞
\
k
R(Ψα−β,β
)=
k=1
(
=
∞
\
k
R(Φα−β
)
k=1
S(Rd ),
S
γ∈[α−β,2] Sγ
(Rd ) ∩ C
α−β (R
d
),
when α − β ≤ 0
when 0 < α − β < 1,
by the assumption of induction and Lemma 3.1. When α − β ≤ 0, it follows from (3.5) that for
m ∈ N,
m
Φαm S(Rd ) ∩ D(Φαm ) ⊂ R(Ψα,β
) ⊂ R(Φαm ),
which yields (3.6) by Lemma 3.1. When 0 < α − β < 1, it follows from (3.5) that for m ∈ N,
[
m
) ⊂ R(Φαm ),
(3.7)
Sγ (Rd ) ∩ D(Φαm ) ⊂ R(Ψα,β
Φαm
γ∈[α−β,2]
since D(Φαm ) ⊂ D(Φα ) ⊂ Iα (Rd ) ⊂ Iα−β (Rd ) ⊂ Cα−β (Rd ). Thus Lemma 3.1 yields (3.6). Then
(3.6) is true for all n ∈ N, namely, (3.4) holds for all α < 1.
We next show
∞
\
∞
\
m
R(Ψα,β
)=
m=1
m=1
R(Φαm ) for all α ∈ 1 + (n − 1)β, 1 + nβ ∩ (−∞, 2),
(3.8)
for each n ∈ Z+ by induction. If n = 0, then α < 1 and we have just shown the case.
Assume that
(3.8) holds for n − 1 in place of n with n ≥ 1. Then
for
α
∈
1
+
(n
−
1)β,
1
+
nβ
∩ (−∞, 2), it
follows that α − β ∈ 1 + (n − 2)β, 1 + (n − 1)β ∩ (−∞, 2). Then the assumption of induction
and Lemma 3.1 yields that
∞
\
k
R(Ψα−β,β
)=
k=1
=
∞
\
k
R(Φα−β
)
k=1
S(Rd ),
S
Sγ∈[α−β,2]
Sγ
(Rd ) ∩ C
α−β (R
d
),
d
0
d
γ∈[α−β,2] Sγ (R ) ∩ Cα−β (R ),
when α − β ≤ 0
when 0 < α − β < 1,
when 1 < α − β < 2.
When α − β < 1, (3.8) holds by the same argument as above. When 1 < α − β < 2, the same
0
0
(Rd ).
(Rd ) ⊂ Cα−β
inclusion as (3.7) follows from (3.5), since D(Φαm ) ⊂ D(Φα ) ⊂ Iα0 (Rd ) ⊂ Iα−β
Therefore Lemma 3.1 yields (3.8). Then (3.8) is true for all n ∈ Z+ . Thus (3.4) holds for all
α ∈ (1, 2) \ {1 + nβ : n ∈ N}.
We finally prove Theorem 2.6.
Proof of Theorem 2.6. (3.5) with m = 1 yields that R(Ψα,β ) ⊂ R(Φα ). Also we have
∞
\
m=1
R(Φαm ) = R(Φα ) ∩ S(Rd ),
Compositions of mappings of infinitely divisible distributions
237
which is Theorem 5.2 of Maejima and Ueda [19]. Then we have
R(Ψα,β ) ∩ S(Rd ) ⊂ R(Φα ) ∩ S(Rd ) =
∞
\
R(Φαm ) =
m=1
∞
\
m
R(Ψα,β
),
m=1
where the last equality follows from Theorem 3.2. Furthermore, we have that
∞
\
m=1
and that
∞
\
m=1
Thus
T∞
m
m=1 R(Ψα,β )
m
R(Ψα,β
)=
m
R(Ψα,β
) ⊂ R(Ψα,β ),
∞
\
m=1
R(Φαm ) = R(Φα ) ∩ S(Rd ) ⊂ S(Rd ).
⊂ R(Ψα,β ) ∩ S(Rd ). Therefore (2.1) holds.
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