A Class of Infinitely Divisible Multivariate and Matrix Gamma Distributions and Cone-valued Generalised Gamma Convolutions

0.44 MB·English

by  Victor Pérez-Abreu

A Class of Infinitely Divisible Multivariate and Matrix
Gamma Distributions and Cone-valued Generalised
Gamma Convolutions
Victor Pérez-Abreu∗ Robert Stelzer†
2
1 Abstract
0
ClassesofmultivariateandconevaluedinfinitelydivisibleGammadistributionsare
2
introduced. Particularemphasisisputonthecone-valuedcase,duetotherelevance
n
of infinitely divisible distributions on the positive semi-definite matrices in applica-
a
J tions. Thecone-valuedclassofgeneralisedGammaconvolutionsisstudied. Inpartic-
ular,acharacterisationintermsofanItô-Wienerintegralwithrespecttoaninfinitely
6
divisiblerandommeasureassociatedtothejumpsofaLévyprocessisestablished.

] AnewexampleofaninfinitelydivisiblepositivedefiniteGammarandommatrixis
R
introduced. Ithaspropertieswhichmakeitappealingformodellingunderaninfinite
P divisibility framework. An interesting relation of the moments of the Lévy measure
h. andtheWishartdistributionishighlightedwhichwesupposetobeimportantwhen
t consideringthelimitingdistributionoftheeigenvalues.
a
m
Keywords: infinitedivisibility;randommatrix;conevalueddistribution;Lévyprocess;matrix
[
subordinator.

1 AMS2010SubjectClassification: Primary60E07;60G51,Secondary60B20;60G60.
v
1
6
1 Introduction
4
1
TheclassicalexamplesofmultivariateandmatrixGammadistributionsintheprob-
.
1 ability and statistics literature are not necessarily infinitely divisible [14], [19], [40].
0
These examples are analogous to one-dimensional Gamma distributions and are ob-
2
tained by a direct generalisation of the one-dimensional probability densities; see for
1
: example [15], [23], [24]. Working in the domain of Fourier transforms, some infinitely
v
divisible matrix Gamma distributions have recently been considered in [5], [27]. Their
i
X
Lévy measures are direct generalisations of the one-dimensional Gamma distribution.
r The work of [27] arose in the context of random matrix models relating classical and
a
freeinfinitelydivisibledistributions.
The study of infinitely divisible random elements in cones has been considered in
[4], [25], [26], [31]andreferencestherein. Theyareimportantintheconstructionand
modelingofconeincreasingLévyprocesses. Intheparticularcaseofinfinitelydivisible
positive-definite random matrices, their importance in applications has been recently
highlighted in [7], [8], [28] and [29]. This is due to the fact that infinite divisibility
allowsmodellingbymatrixLévyandOrnstein-Uhlenbeckprocesses,whichareinthose
∗DepartmentofProbabilityandStatistics,CenterforResearchinMathematicsCIMAT,Apdo. Postal402,
Guanajuato,Gto.36000México,Email: [email protected]
†InstituteofMathematicalFinance,UlmUniversity,Helmholtzstraße18,D-89069Ulm,Germany. Email:
[email protected],http://www.uni-ulm.de/mawi/finmath
2 InfinitelyDivisibleMultivariateGammaDistributions
papersusedtomodelthetimedynamicsofad×dcovariancematrixtoobtainaso-called
stochasticvolatilitymodel(forobservedseriesoffinancialdata).
GeneralizedGammaConvolutions(GGC)isarichandinterestingclassofone-dimen-
sional infinitely divisible distributions on the cone R+ = [0,∞). It is the smallest class
ofinfinitelydivisibledistributionsonR+ thatcontainsallGammadistributionsandthat
isclosedunderclassicalconvolutionandweakconvergence. Thisclasswasintroduced
byO.Thorinin aseriesofpapersandfurtherstudiedbyL.Bondessoninhisbook[10].
The book of Steutel and Van Harn [39] contains also many results and examples about
GGC.SeveralwellknownandimportantdistributionsonR+areGGC.Therecentsurvey
paper by James, Roynette and Yor [16] contains a number of classical results and old
andnewexamplesofGGC.ThemultivariatecasewasconsideredinBarndorff-Nielsen,
MaejimaandSato[3].
There are three main purposes in this paper. We formulate and study multivariate
and cone valued Gamma distributions which are infinitely divisible. Second, we con-
sider and characterise the corresponding class GGC(K) of Generalised Gamma Con-
volutions on a finite dimensional cone K. Finally, we introduce a new example of a
positive definite random matrix with infinitely divisible Gamma distribution and with
explicitLévymeasure.
The main results and organisation of the paper are as follows. Section 2 briefly
presents preliminaries on notation and results about one-dimensional GGC on R+ as
wellassomematrixnotation. Section3introducesaclassofinfinitelydivisibled-variate
GammadistributionsΓd(α,β),whoseLévymeasuresareanalogoustotheLévymeasure
oftheone-dimensionalGammadistribution. Theparametersαandβ aremeasuresand
functions on S (the unit sphere with respect to a prescribed norm), respectively. It is
shown that the distribution does not depend on the particular norm under consider-
ation. The characteristic function is derived and it is shown that the Fourier-Laplace
transformonCd existsifβ isboundedawayfromzeroα−almosteverywhere. Further-
more,thefinitenessofmomentsofallordersisstudiedandsomeinterestingexamples
exhibitingessentialdifferencestounivariateGammadistributionsaregiven.
Section4considersconevaluedGammadistributionsandtheircorrespondingclass
GGC(K)ofGeneralisedGammaConvolutionsonaconeK,definedasthesmallestclass
ofdistributionsonK whichisclosedunderconvolutionandweakconvergenceandcon-
tains all the so-called elementary Gamma variables in K (and also all Gamma random
variablesinK inournewdefinition). Thisclassischaracterisedasthestochasticinte-
gralofanon-randomfunctionwithrespecttothePoissonrandommeasureofthejumps
ofaGammaLévyprocessonthecone. Thisisanewrepresentationinthemultivariate
case extending the Wiener-Gamma integral characterization of one-dimensional GGC
onR+ =[0,∞),asconsidered,forexample,in[16].
Section 5 considers the special cone valued case of infinitely divisible positive-
semidefinite d×d matrix Gamma distributions. New examples are introduced via an
explicitformoftheirLévymeasure. Theyincludeasparticularcasestheexamplescon-
sideredin[5],[27]. Adetailedstudyisdoneofthenewtwoparameterpositivedefinite
matrix distribution AΓ(η,Σ), where η > (d−1)/2 and Σ is a d×d positive definite ma-
trix. This special infinitely divisible Gamma matrix distribution has several modeling
featuressimilartotheclassical(butnon-infinitelydivisible)matrixGammadistribution
definedthroughadensity,inparticulartheWishartdistribution. Namely,momentsofall
ordersexist,thematrixmeanisproportionaltoΣandthematrixofcovariancesequals
the second moment of the Wishart distribution. When Σ is the d×d identity matrix
Id, the distribution is invariant under orthogonal conjugations and the trace of a ran-
dom matrix M with distribution AΓ(η,Id) has a one-dimensional Gamma distribution.
InfinitelyDivisibleMultivariateGammaDistributions 3
A relation of the moments of the Marchenko-Pastur distribution with the asymptotic
momentsoftheLévymeasureisexhibited. Hence,thismatrixGammadistributionhas
a special role when dealing with a random covariance matrix and its time dynamics,
e.g. byspecifyingitasamatrixLévyorOrnstein-Uhlenbeckprocess. Asanapplication,
thematrixNormal-Gammadistributionisintroduced,whichisamatrixextensionofthe
one-dimensionalvarianceGammadistributionof[22]whichispopularinfinance.
2 Preliminaries
For the general background in infinitely divisible distributions and Lévy processes
werefertothestandardreferences,e.g. [36].
2.1 One-dimensional GGC
ApositiverandomvariableY withlawµ=L(Y)belongstotheclassofGeneralised
Gamma Convolutions (GGC) on R+ = [0,∞), denoted by T(R+), if and only if there
existsapositiveRadonmeasureυµ on(0,∞)anda>0suchthatitsLaplacetransform
isgivenby:
(cid:18) (cid:90) ∞ (cid:16) z(cid:17) (cid:19)
Lµ(z)=Ee−zY =exp −az− ln 1+ s υµ(ds) (2.1)
0
with
(cid:90) 1 (cid:90) ∞ υ (dx)
|logx|υµ(dx)<∞, µx <∞. (2.2)
0 1
For convenience we shall work without the translation term, i.e. with a = 0. The mea-
sure υµ is called the Thorin measure of µ. Its Lévy measure is concentrated on (0,∞)
andissuchthat:
νµ(dx)=x−1lµ(x)dx, (2.3)
wherelµ isacompletelymonotonefunctioninx>0givenby
(cid:90) ∞
lµ(dx)= e−xsυµ(ds). (2.4)
0
The class T(R+) can be characterized by Wiener-Gamma representations. Specif-
ically, a positive random variable Y belongs to T(R+) if and only if there is a Borel
functionh:R+ →R+ with
(cid:90) ∞
ln(1+h(t))dt<∞, (2.5)
0
suchthatY =L Yh hastheWiener-Gammaintegralrepresentation
(cid:90) ∞
Yh =L h(u)dγu, (2.6)
0
where(γt;t≥0)isthestandardGammaprocesswithLévymeasureν(dx)=e−xdxx.The
relation between the Thorin function h and the Thorin measure υµ is as follows: υµ is
the image of the Lebesgue measure on (0,∞) under the application : s → 1/h(s). That
is,
(cid:90) ∞ (cid:90) ∞
e−h(xs)ds= e−xzυµ(dz), x>0. (2.7)
0 0
Ontheotherhand,ifFυµ(x)=(cid:82)0xυµ(dy)forx≥0andFυ−µ1(s)isthetherightcontinuous
generalised inverse of Fυµ(s), that is Fυ−µ1(s) = inf{t > 0;Fυµ(t) ≥ s} for s ≥ 0, then,
h(s)=1/F−1(s)fors≥0.
υµ
4 InfinitelyDivisibleMultivariateGammaDistributions
ManywellknowndistributionsbelongtoT(R+). Thepositiveα-stabledistributions,
0 < α < 1, are GGC with h(s) = {sθΓ(α + 1)}−α1 for a θ > 0. In particular, for the
1/2−stable distribution, h(s) = 4(cid:0)s2π(cid:1)−1. Beta distribution of the second kind, lognor-
malandParetoarealsoGGC,see[16].
FormoredetailsonunivariateGGCswereferto[10,16]
2.2 Notation
Md(R) is the linear space of d×d matrices with real entries and Sd its subspace
of symmetric matrices. By S+d and S+d we denote the open (in Sd) and closed cones of
positive and nonnegative definite matrices in Md(R). SRd,(cid:107)·(cid:107) is the unit sphere on Rd
withrespecttothenorm(cid:107)·(cid:107).
TheFouriertransformµ(cid:98)ofameasureµonM=Rd orM=Md(R)isgivenby
(cid:90)
µˆ(z)= ei(cid:104)z,x(cid:105)µ(dx) z ∈M
M
where we use (cid:104)A,B(cid:105) = tr(A(cid:62)B) as the scalar product in the matrix case, where A(cid:62)
denotes the transposed on Md(R). By Id we denote the d×d identity matrix and by |A|
the determinant of a square matrix A. For a matrix A in the linear group GLd(R) we
writeA−(cid:62) =(cid:0)A(cid:62)(cid:1)−1.
Wesaythatthedistributionofasymmetricrandomd×dmatrixM isinvariantunder
orthogonal conjugations if the distribution of OMO(cid:62) equals the distribution of M for
any non-random matrix O in the orthogonal group O(d). Note that M → OMO(cid:62) with
O ∈O(d)arealllinearorthogonalmapsonS+d (orMd)preservingS+d.
3 Multivariate Gamma Distributions
3.1 Definition
Definition 3.1.Let µ be an infinitely divisible probability distribution on Rd. If there
existsafinitemeasureαontheunitsphereSRd,(cid:107)·(cid:107)withrespecttothenorm(cid:107)·(cid:107)equipped
withtheBorelσ-algebraandaBorel-measurablefunctionβ :SRd,(cid:107)·(cid:107) →R+ suchthat
(cid:32) (cid:33)
(cid:90) (cid:90) (cid:16) (cid:17)e−β(v)r
µˆ(z)=exp eirv(cid:62)z−1 drα(dv) (3.1)
r
SRd,(cid:107)·(cid:107) R+
for all z ∈ Rd, then µ is called a d-dimensional Gamma distribution with parameters α
andβ,abbreviatedΓd(α,β)-distribution.
Ifβ isconstant,wecallµa (cid:107)·(cid:107)-homogeneousΓd(α,β)-distribution.
Observe that the notation Γd(α,β) implicitly also specifies which norm we use, be-
cause α is a measure on the unit sphere with respect to the norm employed and β is
a function on it. The parameters α and β play a comparable role as shape and scale
parametersasintheusualpositiveunivariatecase.
Remark 3.2.(i)ObviouslytheLévymeasureνµ ofµisgivenby
(cid:90) (cid:90) e−β(v)r
νµ(E)= 1E(rv) r drα(dv) (3.2)
SRd,(cid:107)·(cid:107) R+
forallE ∈B(Rd). Thisexpressionisequivalentto
e−β(x/(cid:107)x(cid:107))(cid:107)x(cid:107)
νµ(dx)= (cid:107)x(cid:107) α(cid:101)(dx), x∈Rd (3.3)
InfinitelyDivisibleMultivariateGammaDistributions 5
whereα(cid:101) isameasureonRd givenby
(cid:90) (cid:90) ∞
α(cid:101)(E)= 1E(rv)drα(dv), E ∈B(Rd). (3.4)
SRd,(cid:107)·(cid:107) 0
(ii) Likewise we define Md(R) and Sd-valued Gamma distributions with parameters
αandβ (abbreviatedΓMd(α,β)andΓSd(α,β),respectively)byreplacingRd withMd(R)
and Sd, respectively, and the Euclidean scalar product with (cid:104)Z,X(cid:105) =tr(X(cid:62)Z). All up-
comingresultsimmediatelygeneralisetothismatrix-variatesetting. Weprovidefurther
detailsinSection5.
If d = 1 and α({−1}) = 0, then we have the usual one-dimensional Γ(α({1}),β(1))-
distribution. In general it is elementary to see that for d = 1 a random variable X ∼
D
Γ1(α,β)ifandonlyifX =X1−X2withX1 ∼Γ(α({1}),β(1))andX2 ∼Γ(α({−1},β(−1))
being two independent usual Gamma random variables, i.e. X has a bilateral Gamma
distribution as analysed in [17, 18] and introduced in [11, 22] under the name vari-
ance Gamma distribution. If α({1}) = α({−1}) and β(1) = β(−1), it indeed can be
represented as the variance mixture of a normal random variable with an independent
positiveGammaone(acomprehensivesummaryofthiscasecanbefoundin[39]where
itiscalledsym-Gammadistribution).
Nowweaddressthequestionofwhichα,β wecantaketoobtainaGammadistribu-
tion.
Proposition 3.3.Let α be a finite measure on SRd,(cid:107)·(cid:107) and β : SRd,(cid:107)·(cid:107) → R+ a measur-
able function. Then (3.2) defines a Lévy measure νµ and thus there exists a Γd(α,β)
probabilitydistributionµifandonlyif
(cid:90) (cid:18) 1 (cid:19)
ln 1+ α(dv)<∞. (3.5)
β(v)
SRd,(cid:107)·(cid:107)
(cid:82)
Moreover, Rd((cid:107)x(cid:107)∧1)νµ(dx)<∞holdstrue.
The condition (3.5) is trivially satisfied, if β is bounded away from zero α-almost
everywhere.
Proof.
(cid:90) (cid:90) (cid:90) 1 (cid:90) 1−e−β(v)
(cid:107)x(cid:107)ν (dx)= e−β(v)rdrα(dv)= α(dv)
µ β(v)
(cid:107)x(cid:107)≤1 SRd,(cid:107)·(cid:107) 0 SRd,(cid:107)·(cid:107)
≤α(S )<∞
Rd,(cid:107)·(cid:107)
using the elementary inequality 1 − e−x ≤ x,for each x ∈ R+. Denoting by E1 the
exponentialintegralfunctiongivenbyE1(z)=(cid:82)z∞ e−ttdtforz ∈R+,weget
(cid:90) (cid:90) (cid:90) ∞ e−β(v)r (cid:90)
νµ(dx)= r drα(dv)= E1(β(v))α(dv) (3.6)
(cid:107)x(cid:107)>1 SRd,(cid:107)·(cid:107) 1 SRd,(cid:107)·(cid:107)
(cid:90) ∞
= E1(z)τ(dz), (3.7)
0
where we made the substitution z = β(v) and τ(E) = α(β−1(E)) for all Borel sets E in
R+. Sinceτ isafinitemeasureand0≤E1(z)≤e−zln(1+1/z)∀z ∈R+(see[1,p. 229]),
(cid:90) ∞
E (z)τ(dz)<∞.
1
1/2
6 InfinitelyDivisibleMultivariateGammaDistributions
The series representation E1(z) = −γ − ln(z) − (cid:80)∞i=1 (−n1·)ni!zi with γ being the Euler-
Mascheroniconstant([1,p. 229])impliesthatlimz↓0E1(z)/(−ln(z))=1. Hence,
(cid:90) 1/2 (cid:90) 1/2 (cid:90) 1/2
E (z)τ(dz)<∞⇔ |ln(z)|τ(dz)<∞⇔ ln(1+1/z)τ(dz)<∞
1
0 0 0
using ln(1+1/z) = ln(1+z)−ln(z) and the finiteness of τ in the second equivalence.
Appealingtothefinitenessofτ oncemore,theaboveconditionsareequivalentto
(cid:90) ∞ (cid:90)
ln(1+1/z)τ(dz)= ln(1+1/β(v))α(dv)<∞.
0 SRd,(cid:107)·(cid:107)
The next proposition shows that the definition of a Gamma distribution does not
dependonthenorm,onlytheparametrisationchangeswhenusingdifferentnorms.
Proposition3.4.Let(cid:107)·(cid:107)abeanormonRdandµbeaΓd(α,β)distributionwithαbeing
a finite measure on SRd,(cid:107)·(cid:107)a and β : SRd,(cid:107)·(cid:107)a → R+ measurable. If (cid:107)·(cid:107)b is another norm
on Rd, then µ is a Γd(αb,βb) distribution with αb being a finite measure on SRd,(cid:107)·(cid:107)b and
βb :SRd,(cid:107)·(cid:107)b →R+ measurable. Moreover,itholdsthat
(cid:90) (cid:18) v (cid:19)
αb(E)= 1E (cid:107)v(cid:107) α(dv) ∀E ∈B(SRd,(cid:107)·(cid:107)b) (3.8)
SRd,(cid:107)·(cid:107)a b
(cid:18) (cid:19)
v
βb(vb)=β (cid:107)vb(cid:107) (cid:107)vb(cid:107)a ∀vb ∈SRd,(cid:107)·(cid:107)b. (3.9)
b a
The above formulae show that the mass in the different directions, which is given
byα,doesnotchange,andβ onlyneedstobeadaptedforthescalechangesimpliedby
thechangeofthenorm.
Proof. Substitutingfirstvb =v/(cid:107)v(cid:107)b andthens=r/(cid:107)vb(cid:107)a gives:
(cid:32) (cid:33)
(cid:90) (cid:90) (cid:16) (cid:17)e−β(v)r
exp eirv(cid:62)z−1 drα(dv)
r
SRd,(cid:107)·(cid:107)a R+
=exp(cid:90) (cid:90) (cid:16)ei(cid:107)vbr(cid:107)avb(cid:62)z−1(cid:17)e−β(cid:16)(cid:107)rvvbb(cid:107)a(cid:17)rdrαb(dvb)
SRd,(cid:107)·(cid:107)b R+
=exp(cid:90) (cid:90) (cid:16)eisvb(cid:62)z−1(cid:17)e−β(cid:16)(cid:107)vvbb(cid:107)sa(cid:17)(cid:107)vb(cid:107)asdsαb(dvb).
SRd,(cid:107)·(cid:107)b R+
3.2 Properties
InthissectionwestudyseveralfundamentalpropertiesofourGammadistributions.
Proposition 3.5.AnyΓd(α,β)-distributionisself-decomposable.
Proof. Thisfollowsimmediatelyfromthedefinitionand[36,Th. 15.10].
Later on we will considerably improve this result by showing that we are in a very
special subset of the self-decomposable distributions. This result has important impli-
cationsforapplicationswhereonelikestoworkwithdistributionshavingdensities,i.e.
distributionswhichareabsolutelycontinuous(withrespecttotheLebesguemeasure).
InfinitelyDivisibleMultivariateGammaDistributions 7
Proposition 3.6.Assumethatsuppαisoffulldimension,i.e. thatitcontainsdlinearly
independentvectorsinRd. ThentheΓd(α,β)-distributionisabsolutelycontinuous.
Proof. Itisimmediatethatthesupportof Γd(α,β)istheclosedconvexconegenerated
by suppα. Hence, the support of Γd(α,β) is of full dimension and so the distribution is
non-degenerate. Thus[35]concludes.
It follows along the same lines that in the degenerate case the Γd(α,β)-distribution
is absolutely continuous with respect to the Lebesgue measure on the subspace gen-
erated by suppα. If suppα consists of exactly d linearly independent vectors, Γd(α,β)
equalsthedistributionofalineartransformationofavectorofdindependentunivariate
Gammarandomvariableswithappropriateparametersandthusthedensitycanbecal-
culated easily using the density transformation theorem with an invertible linear map.
Ifsuppαisafinitesetoffulldimension,onecancalculatethedensityfromthedensityof
independent univariate Gamma random variables by using the density transformation
theoremwithaninvertiblelinearmapandintegratingoutthenon-relevantdimensions.
Ingeneralthedensitycanbedeterminedviasolvingapartialintegro-differentialequa-
tion(see[37]). Moreover,criteriaforqualitativepropertiesofthedensitylikecontinuity
and continuous differentiability can be deduced from the results of [33, 34], but look-
ing at the simple case of a vector of independent univariate Gamma distributions one
immediately sees that the sufficient conditions given there are far from being sharp.
Thereforewerefrainfromgivingmoredetails.
Next we show that our d-dimensional Gamma distribution has the same closedness
propertiesregardingscalingandconvolutionastheusualunivariateone.
Proposition 3.7.(i)LetX ∼Γd(α,β)andc>0. ThencX ∼Γd(α,β/c).
(ii)LetX1 ∼Γd(α1,β)andX2 ∼Γd(α2,β)betwoindependentd-dimensionalGamma
variables. ThenX1+X2 ∼Γd(α1+α2,β).
Proof. Followsimmediatelyfromconsideringthecharacteristicfunctions.
Likewiseitisimmediatetoseethefollowingdistributionalpropertiesoftheinduced
Lévyprocess.
Proposition 3.8.Let L be a Γd(α,β) Lévy process, i.e. L1 ∼ Γd(α,β). Then Lt ∼
Γd(tα,β)forallt∈R+.
Of high importance for applications is that the class of Γd distributions is invariant
underinvertiblelineartransformations.
Proposition3.9.LetX ∼Γd(α,β)(withrespecttothenorm(cid:107)·(cid:107))andAbeaninvertible
d×dmatrix. ThenAX ∼Γd(αA,βA)withrespecttothenorm(cid:107)·(cid:107)A =(cid:107)A−1·(cid:107)and
(cid:90)
αA(E)= 1E(Av)α(dv) =α(A−1E) ∀E ∈B(SRd,(cid:107)·(cid:107)A) (3.10)
SRd,(cid:107)·(cid:107)
βA(v)=β(cid:0)A−1v(cid:1) ∀v ∈SRd,(cid:107)·(cid:107)A. (3.11)
Proof. Wehaveforallz ∈Rd
E(cid:0)ei<z,AX>(cid:1)=(cid:90) ei<z,Ax>µ(dx)=(cid:90) (cid:90) (cid:16)eirv(cid:62)A(cid:62)z−1(cid:17)e−β(v)rdrα(dv)
r
Rd SRd,(cid:107)·(cid:107) R+
(cid:90) (cid:90) (cid:16) (cid:17)e−β(A−1u)r
= eiru(cid:62)z−1 drα(A−1du)
r
SRd,(cid:107)·(cid:107)A R+
=(cid:90) (cid:90) (cid:16)eiru(cid:62)z−1(cid:17)e−βA(u)rdrα (du)
r A
SRd,(cid:107)·(cid:107)A R+
8 InfinitelyDivisibleMultivariateGammaDistributions
wherewesubstitutedu=Av.
It is easy to see that the above proposition can be extended to m×d matrices of
full rank with m > d. Obviously, such a result cannot hold in general for a linear
transformation A with ker(A) (cid:54)= {0}, since combinations of one dimensional Gamma
distributionsareingeneralnotunivariateGammadistributions.
Nextwepresentanalternativerepresentationofthecharacteristicfunction.
Proposition 3.10.Let µ be Γd(α,β) distributed. Then the characteristic function is
givenby
(cid:32)(cid:90) (cid:18) β(v) (cid:19) (cid:33)
µˆ(z)=exp ln α(dv) forall z ∈Rd (3.12)
β(v)−iv(cid:62)z
SRd,(cid:107)·(cid:107)
wherelnisthemainbranchofthecomplexlogarithm.
Proof. Followsfromthedefinitionandthewellknownfact
(cid:90) ∞(cid:16) (cid:17)e−β(v)r (cid:18) β(v) (cid:19)
e−r(−iv(cid:62))z−1 dr =ln .
r β(v)−iv(cid:62)z
0
Notethatifαhascountablesupport{vj}j∈N,then
(cid:32) (cid:33)α({vj})
µˆ(z)= (cid:89) β(vj) .
β(v )−iv(cid:62)z
j∈N j j
We now show that the Fourier-Laplace transform of a Gamma distribution exists if
andtoacertainextentonlyifβ isboundedawayfromzeroαalmosteverywhere.
Theorem 3.11.(i) The Fourier-Laplace transform µˆ of a Γd(α,β) distribution µ exists
for all z in a neighborhood U ⊆ Cd of zero, if β(v) ≥ κ for v ∈ SRd,(cid:107)·(cid:107) α-a.e. with κ > 0.
µˆ isanalyticthereandgivenbyformula (3.12).
(ii)Ifthereexistsasequence(vn)n∈N inSRd,(cid:107)·(cid:107) withlimn→∞β(vn)=0andα({vn})>
0foralln∈N,thentheFourier-Laplacetransformµˆ existsinnoneighborhoodU ⊆Cd
ofzero.
Proof. Using Proposition 3.4 we can assume w.l.o.g. that the Euclidean norm (cid:107)·(cid:107)2 is
usedforthedefinitionoftheΓd(α,β)distribution.
(i) We will now show (i) for U = Bκ(0) ⊆ Cd, where Bκ(0) := {x ∈ Cd : (cid:107)x(cid:107)2 < κ}.
FromProposition3.10itisclearthatµˆ(z)existsforallz ∈Bκ(0)⊆Cd,ifandonlyif
(cid:90) (cid:18) β(v) (cid:19) (cid:90) (cid:18) iv(cid:62)z(cid:19)
ln α(dv)=− ln 1− α(dv)
β(v)−iv(cid:62)z β(v)
SRd,(cid:107)·|2 SRd,(cid:107)·(cid:107)2
exists for all z ∈ Bκ(0). Consider now an arbitrary δ ∈ (0,1) and z ∈ Bδκ(0). Then
the Cauchy-Schwarz inequality implies |iv(cid:62)z| ≤ (cid:107)z(cid:107)2 ≤ δκ and hence |(iv(cid:62)z)/β(v)| ≤ δ.
(cid:16) (cid:17)
Thereforeln 1− iβv((cid:62)vz) existsandisboundedonBδκ(0)α-a.e. Thisimpliesthat
(cid:90) (cid:18) iv(cid:62)z(cid:19)
− ln 1− α(dv)
β(v)
SRd,(cid:107)·(cid:107)2
existsonBδκ(0). Sinceδ ∈(0,1)wasarbitrary,thisconcludestheproofof(i),sincethe
analyticityfollowsimmediatelyfromtheappendixof[12].
InfinitelyDivisibleMultivariateGammaDistributions 9
(ii)W.l.o.g. assumeβ(vn)<1/n. Forn∈Nsetzn =−iβ(vn)vn. Then(cid:107)zn(cid:107)2 =β(vn)<
1/nand1−(ivn(cid:62)zn)/β(vn)=0. Hence,
(cid:90) (cid:18) iv(cid:62)z (cid:19)
ln 1− n α(dv)
β(v)
{vn}
andthereby
(cid:90) (cid:18) iv(cid:62)z (cid:19)
ln 1− n α(dv)
β(v)
SRd,(cid:107)·(cid:107)2
do not exist. This implies that µˆ is not defined on B (0). Since n ∈ N was arbitrary,
1/n
thisshows(ii).
Remark3.12.Toextendthisresulttothematrixcase,onesimplyhastousetheFrobe-
nius or trace norms and the scalar product Z,X (cid:55)→ tr(X(cid:62)Z) instead of the Euclidean
normandscalarproduct. WeconsiderthisinSection5.
Proposition 3.13.A Γd(α,β) distribution µ has a finite moment of order k > 0, i.e.
(cid:82) (cid:107)x(cid:107)kµ(dx)<∞,ifandonlyif
Rd
(cid:90)
β(v)−kα(dv)<∞. (3.13)
SRd,(cid:107)·(cid:107)
Moreover, if m is the mean vector and Σ = (σij)i,j=1,...,d is the covariance matrix of
Γ (α,β)
d
(cid:90)
m= β(v)−1vα(dv). (3.14)
SRd,(cid:107)·(cid:107)
and
(cid:90)
Σ= β(v)−2vv(cid:62)α(dv) (3.15)
SRd,(cid:107)·(cid:107)
Proof. If β is bounded away from zero, (3.13) holds trivially and Theorem 3.11 implies
that µ has finite moments of all orders k > 0. So w.l.o.g. assume that β is not bounded
awayfromzerointhefollowing. By[36,p. 162]µhasafinitemomentoforderk,ifand
onlyif
(cid:90) (cid:90) ∞ e−β(v)r
rk drα(dv)<∞.
r
SRd,(cid:107)·(cid:107) 1
Substitutings=rβ(v)thisisequivalentto
(cid:90) (cid:90) ∞
β(v)−k sk−1e−sdsα(dv)<∞. (3.16)
SRd,(cid:107)·(cid:107) β(v)
Assumingwithoutlossofgeneralitythatβ(v)≤1forallv ∈SRd,(cid:107)·(cid:107),wehavethat
(cid:90) ∞ (cid:90) ∞
0<C(k):= sk−1e−sds≤ sk−1e−sds≤Γ(k).
1 β(v)
Hence, (3.16) is equivalent to (3.13). Finally, (3.14) and (3.15) follow from Example
25.12 in [36] and observing that that the infinitely divisible distribution Γd(α,β) with
(cid:82)
Fouriertransform(3.1)hasLévytriplet(ζ,0,νµ),whereζ = (cid:107)x(cid:107)≤1xνµ(dx).
Corollary 3.14.A (cid:107)·(cid:107)-homogeneous Γd(α,β) distribution has an analytic Fourier-La-
placetransforminBβ(0)andfinitemomentsofallorders.
10 InfinitelyDivisibleMultivariateGammaDistributions
Hence,anyhomogeneousGammadistributionbehaveslikeonewouldexpectitfrom
the univariate case. However, the behaviour in the non-homogeneous case may be
drasticallydifferent,asthefollowingexamplesillustrate.
Example 3.15.Considerd=2. Letαbeconcentratedon{vn}n∈N with
v =(sin(n−1),cos(n−1))
n
and set α({vn}) = e−n and β(vn) = 1/n for all n ∈ N. Then by Theorem 3.11 (ii) the
Fourier-Laplacetransformexistsinnoneighbourhoodofzero.
(cid:82)SRd,(cid:107)·(cid:107)2 β(v)−kα(dv)=(cid:80)n∈Nnke−n isfiniteforallk >0usingthequotientcriterion,
because
(n+1)ke−(n+1)
lim =e−1 <1.
n→∞ nke−n
Thus,wehavemomentsofallorders,buttheFourier-Laplacetransformexistsinno
complexneighbourhoodofzero.
Example3.16.Considertheset-upofExample3.15,butsetnowα({vn})=1/n1+m for
somerealm>0. (cid:82)SRd,(cid:107)·(cid:107)2 β(v)−kα(dv)=(cid:80)n∈N n1n+km isfiniteifandonlyifk <m.
Itiseasytoseethatcondition(3.5)issatisfiedifcondition(3.13)holdsforsomek >
0. Hence, the Γ2(α,β) distribution exists indeed, but only moments of orders smaller
thanmarefinite.
Example 3.17.Consider again the set-up of Example 3.15. Set now α({vn}) = (ln(1+
n)3(n+1))−1.
(cid:16) (cid:17)
Then (cid:82)SRd,(cid:107)·(cid:107)2 ln 1+ β(1v) α(dv) = (cid:80)n∈N ln(1+n1)2(1+n) < ∞ (see [32, Theorem 3.29]
andthustheΓ2(α,β)distributioniswell-defined.
Yet, (cid:82)SRd,(cid:107)·(cid:107)2 β(v)−kα(dv) = (cid:80)n∈N ln(1+nn)k3(1+n) = ∞ for all real k > 0 and so the
Γ2(α,β)distributionhasnofinitemomentsofpositiveordersatall.
4 Gamma and Generalised Gamma Convolutions on Cones
4.1 Cone-valued infinitely divisible random elements
Wefirstreviewseveralfactsaboutinfinitelydivisibleelementswithvaluesinacone
of a finite dimensional Euclidean space B with norm (cid:107)·(cid:107) and inner product (cid:104)·,·(cid:105). A
nonempty convex set K of B is said to be a cone if λ ≥ 0 and x ∈ K imply λx ∈ K. A
cone is proper if x = 0 whenever xand −xare in K. The dual cone K(cid:48) of K is defined
as K(cid:48) = {y ∈B(cid:48) :(cid:104)y,s(cid:105)≥0foreverys∈K}. A proper cone K induces a partial order
on B by defining x1 ≤K x2 whenever x2−x1 ∈ K for x1 ∈ B and x2 ∈ B. Examples of
properconesareR+,Rd+ =[0,∞)d,S+d andS+d.
A random element X in K is infinitely divisible (ID) if and only if for each integer
p≥1thereexistpindependentidenticallydistributedrandomelementsX1,...,Xp inK
law
suchthatX = X1+...+Xp.AprobabilitymeasureµonK isIDifitisthedistribution
of an ID element in K. It is known (see [38]) that such a distribution µ is concentrated
(cid:82)
on a cone K if and only if its Laplace transform Lµ(Θ) = Kexp(−(cid:104)Θ,x(cid:105))µ(dx) is given
bytheregularLévy-Khintchinerepresentation
(cid:26) (cid:90) (cid:16) (cid:17) (cid:27)
Lµ(Θ)=exp −(cid:104)Θ,Ψ0(cid:105)− 1−e−(cid:104)Θ,x(cid:105) νµ(dx) forallΘ∈K(cid:48), (4.1)
K
whereΨ0 ∈K andtheLévymeasureissuchthatνµ(Kc)=0and
(cid:90)
((cid:107)x(cid:107)∧1)νµ(dx)<∞. (4.2)
K