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
SubmittedNovember 30, 2009, accepted in final formMay 27, 2010 AMS 2000 Subject classification: 60E07
Keywords: infinitely divisible distribution onRd, stochastic integral mapping, composition of map-pings, limit of nested subclasses
Abstract
Let {Xt(µ),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 mappingΦf(µ)by the law ofRa
0 f(t)d X (µ)
t , from D(Φf)which is the totality ofµ∈I(Rd)such that the stochastic
integralRa
0 f(t)d X (µ)
t is definable, into a class of infinitely divisible distributions. Form∈N, let
Φmf be the mtimes composition of Φf itself. Maejima and Sato (2009) proved that the limits
T∞
m=1Φ m f(D(Φ
m
f))are the same for several known f’s. Maejima and Nakahara (2009) introduced
more general f’s. In this paper, the limitsT∞m=1Φm f(D(Φ
m
f))for such general f’s are investigated
by using the idea of compositions of suitable mappings of infinitely divisible distributions.
1
Introduction
LetP(Rd)be the class of all probability distributions onRd. Throughout this paper,L(X)denotes the law of anRd-valued random variableX andµ(z),b z∈Rd, denotes the characteristic function ofµ∈ P(Rd). AlsoI(Rd)denotes the class of all infinitely divisible distributions onRd.Cµ(z),z∈
Rd, denotes the cumulant function ofµ∈I(Rd), that is,Cµ(z)is the unique continuous function satisfyingµ(z) =b eCµ(z) andC
µ(0) =0. Forµ ∈I(Rd)and t >0, we call the distribution with characteristic function µ(z)b t = et Cµ(z) the t-th convolution of µand write µt for it. We use the
Lévy-Khintchine triplet(A,ν,γ)ofµ∈I(Rd)in the sense that Cµ(z) =−2−1〈z,Az〉+i〈γ,z〉
+
Z
Rd
ei〈z,x〉−1−i〈z,x〉(1+|x|2)−1ν(d x), z∈Rd,
where| · |and〈·,·〉are the Euclidean norm and inner product onRd, respectively,Ais a symmetric nonnegative-definite d×d matrix, γ ∈ Rd and ν is a measure (called the Lévy measure) on Rd satisfyingν({0}) =0 andR
Rd(|x|
2
∧1)ν(d x)<∞. When we want to emphasize the Lévy-Khintchine triplet, we writeµ=µ(A,ν,γ).
We use stochastic integrals with respect to Lévy processes{Xt,t≥0} of nonrandom measurable functions f:[0,∞)→ R, which areRt
0 f(s)d Xs,t∈[0,∞). As the definition of stochastic
inte-grals, we adopt the method in Sato[25, 26]. It is known that if f is locally square integrable on [0,∞), thenRt
0 f(s)d Xs,t∈[0,∞), is definable for any Lévy process{Xt}. The improper
stochas-tic integral R0∞f(s)d Xs is defined as the limit in probability ofR0t f(s)d Xs as t → ∞whenever the limit exists. In our definition,{Rt
0 f(s)d Xs,t∈[0,∞)}is an additive process in law, which is
not always càdlàg in t. If we take its càdlàg modification, the convergence ofRt
0f(s)d Xsabove is
equivalent to the almost sure convergence of the modification ast→ ∞. Let {X(tµ),t≥0}stand for a Lévy process onRd withL(X(µ)
1 ) =µ. Using this Lévy process, we
can define a mapping
Φf(µ) =L
Z a
0
f(t)d Xt(µ)
, µ∈D(Φf)⊂I(Rd), (1.1)
for a nonrandom measurable function f:[0,a)→R, where 0<a≤ ∞andD(Φf)is the domain of a mappingΦf that is the class ofµ∈I(Rd)for whichRa
0 f(t)d X (µ)
t is definable in the sense
above. Also, D0(Φ
f)denotes the totality of µ ∈ I(Rd)satisfying
Ra
0|Cµ(f(t)z)|d t < ∞ for all
z ∈Rd. For a mapping Φf,R(Φ
f)is its range that is{Φf(µ):µ∈D(Φf)}. When we consider
the composition of two mappings Φf andΦg, denoted by Φg◦Φf, the domain of Φg◦Φf is D(Φ
g◦Φf) ={µ∈I(Rd):µ∈D(Φf)andΦf(µ)∈D(Φg)}. For m∈N,Φmf means themtimes
composition ofΦf itself.
A setH⊂ P(Rd)is said to be closed under type equivalence ifL(X)∈HimpliesL(aX+c)∈H fora>0 andc∈Rd. H ⊂I(Rd)is called completely closed in the strong sense (abbreviated as c.c.s.s.) ifHis closed under type equivalence, convolution, weak convergence andt-th convolution for any t>0.
We list below several known mappings. In the following,Ilog(Rd)denotes the totality ofµ∈I(Rd) satisfyingRRdlog
+
|x|µ(d x)<∞, where log+|x|= (log|x|)∨0. (1) U-mapping (Alf and O’Connor[1], Jurek[8]): Let
U(µ) =L
Z 1
0
t d X(tµ)
!
, µ∈D(U) =I(Rd),
(2) Φ-mapping (Wolfe[30], Jurek and Vervaat[15], Sato and Yamazato[29]): Let
Φ(µ) =L
Z ∞
0
e−td X(tµ)
, µ∈D(Φ) =Ilog(Rd),
and let L(Rd)be the class of selfdecomposable distributions onRd. ThenL(Rd) = Φ(Ilog(Rd)). (3) Υ-mapping (Barndorff-Nielsen and Thorbjørnsen[6], Barndorff-Nielsen et al.[4]): Let
Υ(µ) =L
Z1
0
log1 td X
(µ) t
!
, µ∈D(Υ) =I(Rd), (1.2)
and letB(Rd)be the Goldie–Steutel–Bondesson class onRd. ThenB(Rd) = Υ(I(Rd)). (4) G-mapping (Maejima and Sato[18]): Lett=h(s) =R∞
s e− u2
du,s>0, and denote its inverse function bys=h∗(t). Let
G(µ) =L
Zpπ/2
0
h∗(t)d X(tµ)
!
, µ∈D(G) =I(Rd),
and letG(Rd)be the class of generalized typeGdistributions onRd. ThenG(Rd) =G(I(Rd)). (5) Ψ-mapping (Barndorff-Nielsen et al.[4]): Lett=e(s) =R∞
s u
−1e−udu,s>0, and denote its
inverse function bys=e∗(t). Let Ψ(µ) =L
Z ∞
0
e∗(t)d X(tµ)
, µ∈D(Ψ) =I
log(Rd),
and letT(Rd)be the Thorin class onRd. ThenT(Rd) = Ψ(I
log(Rd)).
(6) M-mapping (Aoyama et al.[3]): Lett=m(s) =R∞
s u− 1e−u2
du,s>0, and denote its inverse function bys=m∗(t). Let
M(µ) =L
Z ∞
0
m∗(t)d X(tµ)
, µ∈D(M) =I
log(Rd).
We callM(Rd):=M(Ilog(Rd))the classM and it was actually introduced in Aoyama et al.[3]in the symmetric case.
Remark 1.1. Jurek[13]introduced the mapping
K(e)(µ) =L
Z ∞
0
t d X1(µ−)e−t
,
which is the same asΥin (1.2) by the time change of the driving Lévy process. In the same way, it holds that
G(µ) =L
Z ∞
0
t d Xp(µπ/)2 −h(t)
.
Here we also introduce mappings Φα,α <2, (O’Connor [21, 22], Jurek[9, 10, 11], Jurek and
These mappings are introduced first by Sato[27]forβ=1 and later by Maejima and Nakahara
where
Iα(Rd) =
¨
µ∈I(Rd):
Z
Rd
|x|αµ(d x)<∞
«
, forα >0,
Iα0(Rd) =
¨
µ∈Iα(Rd):
Z
Rd
xµ(d x) =0
«
, forα≥1,
I1∗(Rd) =
(
µ=µ(A,ν,γ)∈I10(Rd): lim
T→∞
Z T
1
t−1d t
Z
|x|>t
xν(d x)exists inRd
)
.
SinceΦ0= Φ,Φ−1=U,Ψ−1,1= Υ,Ψ−1,2=G,Ψ0,1= ΨandΨ0,2=M, the mappingsΦαand 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 subclassesLm(Rd),m∈Z+of L(Rd)in the following way:µ∈Lm(Rd)if and only if
for each b > 1, there exists ρb ∈ Lm−1(Rd)such that µ(z) =b µ(bb −1z)ρbb(z), where L0(Rd) :=
L(Rd). Hereafter we denote the closure under weak convergence and convolution of a class H ⊂ P(Rd)by H. For α ∈ (0, 2], let Sα(Rd) be the class of all α-stable distributions on Rd and letS(Rd) =S
α∈(0,2]Sα(Rd). Then, the limit L∞(Rd):=
T∞
m=0Lm(Rd)is known to be equal
toS(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
ν(B) =
Z
(0,2)
Γ(dα)
Z
S
λα(dξ)
Z∞
0
11B(rξ)r−α−1d r, B∈ B(Rd\ {0}),
whereΓis a measure on(0, 2)satisfying
Z
(0,2)
1 α+
1 2−α
Γ(dα)<∞,
andλα is a probability measure onS := {ξ ∈Rd:|ξ| = 1} for each α ∈ (0, 2), and λα(C)is measurable in α∈(0, 2)for everyC ∈ B(S). This Γis uniquely determined by µand thisλα is uniquely determined byµ up toα ofΓ-measure 0. For the case in more general spaces, see Jurek[7]. For a setA∈ B((0, 2)), letLA
∞(Rd)denote the class ofµ∈L∞(Rd)withΓsatisfying Γ ((0, 2)\A) =0. It is also known that for m∈N, R(Φm) = Lm
−1(Rd). Hence
T∞
m=1R(Φ m) =
L∞(Rd) = S(Rd). In Maejima and Sato[18], nested subclasses R(Um), R(Υm), R(Ψm) and
R(Gm),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 mappingsΨα,1,α∈(0, 2) produce smaller classes thanS(Rd)as the limit of iteration. Maejima and Ueda[19]showed that the mappingΦαhas the same iterated limit as that ofΨα,1forα∈(0, 2). Maejima and Ueda[20]
also constructed a mapping producing a larger class thanS(Rd), which is the closure of the class of semi-stable distributions with a fixed span.
The purpose of this paper is to find the limit of the nested subclassesR(Ψm
α,β),m∈N. For that, we start with the composition ofΨα−β,β andΦα, which will be used for characterizing the nested subclassesR(Ψm
2
Results
Forβ >0, let
Kβ(µ) =µβ=L
Z β
0
d X(tµ)
!
, 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◦Φf = Φf◦ Kβ.
Here
D(Kβ◦Φ
f) =D(Φf ◦ Kβ) =D(Φf).
The following result on composition will be a key in the proof of the main theorem, Theorem 2.4.
Theorem 2.2. Letα∈(−∞, 1)∪(1, 2)andβ >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 withβ = 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 this paper is the following theorem on the limits of the nested subclasses R(Ψm
α,β)which is
T∞
m=1R(Ψmα,β).
Theorem 2.4. Letβ >0. Then
∞
\
m=1
R(Ψm α,β) =
L∞(Rd), forα∈(−∞, 0], L(∞α,2)(Rd), forα∈(0, 1),
L(∞α,2)(Rd)∩I10(Rd), forα∈(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 caseα=0,β=1. Furthermore, the caseβ =1,α∈(0, 2)was already proved by Sato[28]. The caseα=0 is found in Aoyama et al.[2].
We also have the following.
Theorem 2.6. Letβ >0andα∈(−∞, 2)\ {1+nβ:n∈Z+}. Then ∞
\
m=1
R(Ψm
3
Proofs
We first prove Theorem 2.2.
Proof of Theorem 2.2. Forµ∈I(Rd), we have 3.4 and 2.17 of Sato[26] yields the finiteness of (3.1). Then we can use Fubini’s theorem and have
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)
Letα∈[0, 1)∪(1, 2). Letµ∈D(Ψα,β). Note that the domainsD(Ψα,β)andD(Φα)are the same and decreasing inα <2 with respect to set inclusion due to Remark to Theorem 2.8 of Sato[27]. Thenµ∈D(Ψα are not greater than (3.1). Take into account thatD(Ψα
,β) =D0(Ψα,β)forα∈(−∞, 1)∪(1, 2) andβ >0 due to Theorem 2.4 of Sato[27]. Thenµ∈D0(Ψ
(3.1). ThereforeΨα−β,β(µ)∈D(Kβ◦Φα)andΦα(µ)∈D(Kβ◦Ψα−β,β). Henceµ∈D(Kβ◦Φα◦
which yields that R
|t x|>1log|t x|ν(d x)< ∞ a.e. t > 0. Henceµ∈ Ilog(R
which yields that R |t x|>1|x|
We need the following lemma in the proof below.
(ii) If0< α <1and H⊃Sβ∈[α,2]Sβ(Rd), then Maejima and Ueda[19]).
n≥2. Then forα <(nβ)∧1, it follows thatα−β <((n−1)β)∧1. Therefore
by the assumption of induction and Lemma 3.1. Whenα−β≤0, it follows from (3.5) that for m∈N, and Lemma 3.1 yields that
∞ inclusion as (3.7) follows from (3.5), sinceD(Φm
α)⊂D(Φα)⊂ Iα0(Rd)⊂ Iα0−β(R
We finally prove Theorem 2.6.
which is Theorem 5.2 of Maejima and Ueda[19]. Then we have
R(Ψα
,β)∩S(Rd)⊂R(Φα)∩S(Rd) = ∞
\
m=1
R(Φm α) =
∞
\
m=1
R(Ψm α,β),
where the last equality follows from Theorem 3.2. Furthermore, we have that ∞
\
m=1
R(Ψm
α,β)⊂R(Ψα,β),
and that
∞
\
m=1
R(Ψm α,β) =
∞
\
m=1
R(Φm
α) =R(Φα)∩S(Rd)⊂S(Rd). ThusT∞m=1R(Ψm
α,β)⊂R(Ψα,β)∩S(Rd). Therefore (2.1) holds.
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