Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

0.63 MB·English

by  Lancelot F. James

Probability Surveys
Vol.5(2008)346–415
ISSN:1549-5787
DOI:10.1214/07-PS118
Generalized Gamma Convolutions,
Dirichlet means, Thorin measures,
with explicit examples
9 ∗
0
0
Lancelot F. James
2 †
The Hong Kong Universityof Science and Technology
n
Dept. of Information Systems, Business Statisticsand Operations Management
a
Clear WaterBay, Kowloon, Hong Kong SAR e-mail:[email protected]
J

2
Bernard Roynette
2

Institut Elie Cartan, Universit´e Henri Poincar´e,
] B.P. 239, 54506 Vandoeuvre les Nancy Cedex, France
R
P
Marc Yor
.
h
Laboratoire de Probabilit´es et Mod`eles Al´eatoires,
t
a Universit´e Paris VI et VII and
m Institut Universitaire de France,
4 place Jussieu - Case 188
[
F - 75252 Paris Cedex 05,France

2
Abstract:
v
• InSection1,wepresentanumberofclassicalresultsconcerningtheGeneralizedGamma
2
Convolution ( : GGC) variables, their Wiener-Gamma representations, and relation
3
withtheDirichletprocesses.
9
3 • ToaGGCvariable,onemayassociateauniqueThorinmeasure.LetGapositiver.v.
. andΓt(G) resp.Γt(1/G) theGeneralizedGammaConvolutionwithThorinmeasure
8
t-timesthelawofG(resp.thelawof1/G).InSection2,wecomparethelawsofΓt(G)
0 andΓt(1/G(cid:0)). (cid:1)
7
0 • In Section 3, we present some old and some new examples of GGC variables, among
: whichthelengthsofexcursionsofBesselprocessesstraddlinganindependentexponen-
v tialtime.
i
X
r AMS2000subjectclassifications:Primary60E07,60E10,60G51,60G52,
a 60G57.
Keywordsandphrases:Laplacetransform,GeneralizedGammaConvo-
lutions(GGC),WienerGammarepresentation,Stieltjestransform,Dirich-
letprocesses.
ReceivedAugust2007.
∗Thisisanoriginalsurveypaper
†SupportedinpartbygrantsHIA05/06.BM03,RGC-HKUST6159/02P,DAG04/05.BM56
andRGC-HKUST600907oftheHKSAR.
346
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 347
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
1 Classical results on GGC r.v.’s . . . . . . . . . . . . . . . . . . . . . . 349
1.1 The Gamma process . . . . . . . . . . . . . . . . . . . . . . . . . 349
1.2 Wiener-Gamma integrals and GGC variables . . . . . . . . . . . 350
1.3 m-Wiener-Gamma integrals, (m,G) GGC r.v. . . . . . . . . . . . 353
1.4 m-Wiener Gamma integrals, m Dirichlet means, Gamma (m)
mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
1.5 The subordinators Γ (G), t 0 . . . . . . . . . . . . . . . . . . 362
t
≥
1.6 Some examples of GGC subordinators . . . . . . . . . . . . . . . 366
(cid:0) (cid:1)
1.7 The subordinator Γ (S ), t 0 (0<β <1) . . . . . . . . . . 371
t β
≥
1.8 Remark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
(cid:0) (cid:1)
2 A duality principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
2.1 The duality theorem . . . . . . . . . . . . . . . . . . . . . . . . . 373
2.2 Proof of Theorem 2.1. . . . . . . . . . . . . . . . . . . . . . . . . 375
2.3 A complement to the duality theorem . . . . . . . . . . . . . . . 378
2.4 Proof of Theorem 2.3. . . . . . . . . . . . . . . . . . . . . . . . . 379
2.5 Computation of the Thorin measure of a Pareto r.v. . . . . . . . 381
1/α
2.6 Thorin measure associated with γ , 0<α<1 . . . . . . . . . 385
1
3 ExplicitexamplesofGGCvariablesassociatedwiththe(G , 0 α 1)
α
≤ ≤
family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
3.1 The family (G , 0 α 1) . . . . . . . . . . . . . . . . . . . . . 387
α
3.2 Study ofthe subord≤inat≤ors Γ (G ),t 0 and Γ (1/G ),t 0 389
t 1/2 t 1/2
3.3 Study of the r.v.’s Γ (G ) and Γ (≥1/G ), 0<α<1 . .≥. . 396
1 α α 1 α α
3.4 Study of the subordin−ators(cid:0)Γ (µ+G− ), t(cid:1) 0 (cid:0)and Γ 1 , (cid:1)
t 0 ≥ t µ+G0
t 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400
≥ (cid:0) (cid:1) (cid:0) (cid:0) (cid:1)
3.5 Study of the subordinators Γ (G ), t 0 and Γ (1/G ), t 0 403
(cid:1) t 1 ≥ t 1 ≥
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
(cid:0) (cid:1) (cid:0) (cid:1)
Interpolation between the subordinators Γ (1/G), t 0 and (γ , t 0) 407
t t
≥ ≥
List of notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411
(cid:0) (cid:1)
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
Introduction
1. This survey is concerned with the study of a rich and interesting class of
infinitely divisible laws on R called the generalized gamma convolutions
+
(GGC), a class introduced by O. L. Thorin in 1977 see [49] and then studied
thoroughly by L. Bondesson [5]; both the lectures notes by Bondesson and the
(cid:0) (cid:1)
bookbySteutelandVanHarn[48]containmanyresultsonthisclassoflaws.We
shall also discuss their close connections to a class of random variables known
asDirichletmeanswhosestudywasinitiatedbyCifarelliandRegazzini[10;11].
We shall often make, throughout this paper, the common abuse of language
which consists of talking about a random variable instead of its law; thus, we
shall use slightly incorrect terms such as GGC variables, and so on ...
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 348
In order to introduce the family of GGC variables as naturally as possible,
let us consider the 3 sets of r.v.’s (or of laws): , , defined as follows:
G S J
a) is the family of infinitely divisible r.v.’s (taking values in R )
+
b) J is the set of self-decomposable r.v.’s, valued in R
+
S
c) is the set of positive GGC variables
G
The Laplace transforms of these variables satisfy:
E(e λX)=exp aλ ∞(1 e λx)ν(dx) (λ 0)
− −
− − − ≥
(cid:26) Z0 (cid:27)
and
a’) If X , then a 0 and ν(dx) is a L´evy measure, i.e.:
∈J ≥
∞
(1 x)ν(dx)<
∧ ∞
Z0
b’) If X , then a 0 and ν(dx)= dx h(x), with h decreasing
∈S ≥ x
c’) If X , then X and moreover:
∈G ∈S
h(x)= ∞e xyµ(dy)
−
Z0
for a σ-finite and positive measure µ; µ(dy) is called the Thorin measure
of X.
Thus, we have:
G ⊂S ⊂J
and, of course, the inclusions are strict.
Another definition of a GGC variable is sometimes given in the literature
as the limit (in law) of sums of independent gamma variables (with different
parameters).
A comparativediscussionofdifferent(but, in the end, equivalent)properties
ofelements of is made in Jeanblanc,Pitman, andYor [32]. Similarly,it is the
S
aimofthissurveytogatherandtocomparedifferentpropertiesfor .This,and
G
more generally, the entire survey is motivated by the fact that, recently, some
new elements of were discovered; likewise, there is some recurring interest in
G
the Gamma process see, e.g. the Festschrift Volume for Dilip Madan [55] .
(cid:0) (cid:1)
2. This survey paper consists of three parts:
Inthefirstpart,theresultsbeingdiscussedareclassical;theyareaboutthe
•
relationshipsbetweendifferentfamiliesofr.v.’sand/orprocesses,namely:GGC
r.v.’s, Wiener-Gamma integrals,Dirichlet means, compound Poisson processes,
mixing of Gamma variables, Poisson point processes, GGC subordinators, and
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 349
so on. These results are detailed here in order to ease up the reading of this
paper for probabilists coming from diverse horizons.
In the second part, we discuss a notion of duality for the GGC r.v.’s; in
•
particular,whenoneknowsthedensity,orthe LaplacetransformofaGGC r.v.
Γ,thisnotionofdualityallowstocomputeexplicitelythedensity,ortheLaplace
transform of the “dual GGC variable”. We use this principle to compute the
Thorin measure of a Pareto r.v. and of a power of a gamma r.v.
The third part consists essentially in presenting the explicit computations
•
of densities and of Laplace transforms of some particular GGC r.v.’s. In the
main, these r.v.’s originate from the study of the length of excursion which
straddles an independent exponential time for a recurrent Bessel process [4]. It
is noteworthy that, in this third part, the notion of duality presented in the
second part allows to obtain very easily explicit formulae for the density and
the Laplace transform of a large number of GGC variables.
Finally, in the Appendix, we describe an interpolation principle between the
gamma subordinator (γ , t 0) and a family of GGC subordinators.
t
≥
1. Classical results on GGC r.v.’s
A writing convention.
1. Each time we write an equality in law between r.v.’s and that on one or
the other side of this equality, severalr.v.’s occur,we always assume that these
r.v.’s are independent, without mentioning it systematically.
2. It will be convenient, in some instances, to speak of a r.v. instead of its
law and vice versa. We hope that no confusion will ensue.
1.1. The Gamma process
It is a subordinator - i.e. a L´evy process with increasing paths and ca`dla`g tra-
jectories -
Allthroughthis paper,the referenceprocessis the standardGammaprocess
(γ , t 0), which is a subordinator without drift, and with L´evy measure
t
≥
dxe x(x>0). Thus, its L´evy-Khintchine representation writes (see [2]):
x −
E[e−λγt] = exp t ∞(1 e−λx)dxe−x (λ,t 0) (1)
− − x ≥
n Z0 o
1
= exp tlog(1+λ) = (2)
− (1+λ)t
(cid:8) (cid:9)
whereformula(2)isobtainedfrom(1)andfromtheelementaryFrullaniformula
(see [34], p. 6):
∞(1 e−λx)dx ∞e−xzν(dz)= ∞log 1+ λ ν(dz) (3)
− x z
Z0 Z0 Z0 (cid:16) (cid:17)
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 350
where, in general, ν denotes a positive measure on R , which is σ-finite. Here,
+
ν(dz)=δ (dz) is the Dirac measure at 1, but formula (3) shall be useful in the
1
sequel.WenotethatFrullani’sformula(3)maybeeasilyobtainedbyobserving
that the two sides of this formula take the value 0 as λ=0 and have the same
derivative with respect to λ.
For each t>0 fixed, γ follows a gamma law with parameter t:
t
e a
P(γ da)= − at 1 da (a 0) (4)
t −
∈ Γ(t) ≥
This process (γ , t 0) enjoys a large number of remarkable properties which
t
≥
make it a “worthy companion” of Brownian motion. In particular, Emery and
Yor [18] establish a parallel between Brownian motion and its bridges on one
hand, and the Gamma process and its bridges on the other hand. See also
Vershik, Yor and Tsilevich [50] and Yor [55] for a survey of many remarkable
properties of the gamma process.
1.2. Wiener-Gamma integrals and GGC variables
1.2.a Many times throughout this work, we shall use the properties of the
integrals:
∞
Γ(h):= h(s)dγ (5)
s
Z0
where h:R+ R+ is a Boreel function such that:
−→
∞
log 1+h(u) du< (6)
∞
Z0
(cid:0) (cid:1)
Under this hypothesis (6), Γ(h) is finite a.s. (see Proposition 1.1 below). Of
course, since the trajectories of the process are a.s. increasing, the integral fea-
turedin(1.5)maybedefinedeinapath-wisemanner,asausualStieltjesintegral.
We call Γ(h) a Wiener-Gamma integral, in analogy with the Wiener integrals
0∞f(u)dBu,f ∈ L2(R+,du) which constitute the Gaussian space generated
by Browenian motion (B , u 0).
u
R ≥
Thus, the family of r.v.’s Γ(h) with h such that ∞log 1+h(u) du<
0 ∞
constitutes the analoguefor the process(γ , t 0)ofthe firstWiener chaosfor
t
(cid:8) ≥ R (cid:0) (cid:1) (cid:9)
a Brownian motion (B , u e0).
u
≥
Let us assume for a moment that, in (1.5), the function h is constant on a
finite number of intervals, i.e.:
n 1
−
h(s)= h 1 (s) (h 0)
i ]si,si+1] i ≥
i=0
X
for a subdivision 0=s <s < <s < . Then:
0 1 n
··· ∞
n 1
−
Γ(h)= h (γ γ )
i si+1 − si
i=0
X
e
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 351
Thus, Γ(h) is a linear combinationof independent gamma r.v.’s and we obtain:
n 1
e −
E e−λΓ(h) = E e−λhi(γsi+1−γsi)) (λ 0)
≥
i=0
(cid:0) (cid:1) Y (cid:0)
e n 1
− 1
= from(1.2)
i=0 (1+λhi)si+1−si
Y (cid:0) (cid:1)
n 1
= exp − (si+1 si) ∞(1 e−λhix)dxe−x
−i=0 − Z0 − x !
X
from Frullani’s formula (1.3)
= exp (cid:0)∞(1 e−λy)dy ∞e−h(ys)ds(cid:1)
− − y
(cid:18) Z0 Z0 (cid:19)
(after making the change of variableh x=y)
i
= exp ∞(1 e−λy)dy ∞e−yxµh(dx)
− − y
(cid:18) Z0 Z0 (cid:19)
whereµ istheimageofLebesgue’smeasureonR bytheapplications 1
h + → h(s)·
This latter formula justifies the following definition:
1.2.b Definition 1.0: Following ([5], p. 29), we say that a positive r.v. Γ is
a generalized gamma convolution (GGC) - without translation term - if there
exists a positive Radon measure µ on ]0, [ such that:
∞
E[e λΓ] = exp ∞(1 e λx)dx ∞e xzµ(dz) (7)
− − −
− − x
n Z0 Z0 o
∞ λ
= exp log 1+ µ(dz) (8)
− z
n Z0 (cid:16) (cid:17) o
µ(dx)
with: logx µ(dx)< and < (9)
| | ∞ x ∞
Z]0,1] Z[1,∞[
The measure µ is called Thorin’s measure associated with Γ. Thus, from the
L´evy-Khintchine formula, a GGC r.v. is infinitely divisible. In fact, since its
L´evy density lΓ(x)= x1 0∞e−xzµ(dz) satisfies: x−→xl(x) is decreasing, then
Γ is a self decomposable r.v. see, e.g. [37] . Such a self-decomposable r.v. Γ,
R
assumed to be non degenerate, admits a density f such that f (x) > 0 for
Γ Γ
(cid:0) (cid:1)
every x > 0 see [45] p.404 . The study of GGC variables was initiated by O.
Thorin in a series of papers see for instance [49] .
(cid:0) (cid:1)
1.2.c GGC variables and(cid:0)Wiener-Gamma repr(cid:1)esentations.
ThefollowingPropositionisclassical.ThereadermayrefertoLijoiandRegazz-
ini [36].
Proposition 1.1 The class of positive GGC variables coincides with the class
of Wiener-Gamma integrals. More precisely:
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 352
1. If Γ(h)= 0∞h(u)dγu, then:
e R E(e λΓ(h))=exp ∞log 1+ λ µ (dx) (10)
− h
− x
n Z0 (cid:16) (cid:17) o
where µ denotes the imeage of Lebesgue’s measureon R underthe application:
h +
s 1 In other terms:
−→ h(s)·
∞e−h(xs)ds= ∞e−xzµh(dz) (x>0) (11)
Z0 Z0
We note that, in (11), h may vanish on some measurable set.
2. Let Γ denote a GGC r.v. with Thorin measure µ. Let F (x) := µ(dy)
µ ]0,x]
(x 0) and denote by F 1 its right continuous inverse, in the sense of the
≥ µ− R
composition of functions.
Then:
(law) 1
Γ = Γ(h), with h(u)= (12)
Fµ−1(u)
e
Proof of Proposition 1.1 Let Γ(h) := 0∞h(u)dγu. It is easily obtained,
by approximation of Γ by Riemann sums, using also the fact that the L´evy
h R
measure of the process (γ , t 0)eequals dx e x, that:
t ≥ x −
e
E(e λΓ(h)) = exp ∞du ∞(1 e λxh(u))dxe x
− − −
− − x
n Z0 Z0 o
e = exp ∞(1 e−λy)dy ∞e−h(yu)du (13)
− − y
n Z0 Z0 o
after makingthe changeof variablexh(u)=y.We observe,from(11) and(12),
the equivalence of the conditions:
∞ 1
log 1+h(u) du< log 1+ µ (dx)<
h
∞⇔ x ∞
Z0 (cid:0) (cid:1) Z (cid:16) (cid:17) µ (dx)
h
logx µ (dx)< and <
h
⇔ | | ∞ x ∞
Z]0,1] Z[1,∞[
Remark 1.2
1. Formula (13) may be obtained in a slightly different manner: the process
(γ γ := e , s 0) of jumps of the subordinator (γ , t 0) is a Poisson
s − s− s ≥ t ≥
pointprocess whoseintensity measuren equalsthe L´evymeasure of(γ , t 0)
t
≥
see [2] :
1
(cid:0) (cid:1) n(dx)= e−xdx (14)
x
Thus, from the exponential formula for Poisson point processes [43], p. 476 :
t (cid:0) (cid:1)
E exp λ f(s,e ) =exp ds ∞(1 e λf(s,u))n(du)
s −
− − −
h 0<Xs≤t i n Z0 Z0 o
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 353
one gets:
∞
E[exp λ h(s)dγs] = E(exp λ Σ h(s)(∆γs))
− Z0 − s>0
= exp ∞ds ∞(1 e λh(s)u)e−u du ,
−
− − u
(cid:26) Z0 Z0 (cid:27)
which agrees with the expression in (13).
2. Let h,k : R R two Borel functions which satisfy (6) and assume
+ +
−→
(law)
that Γ(h) = Γ(k). Relation (13) and the uniqueness of the L´evy measure in
the L´evy-Khintchinerepresentationimply:the imagesbyh andk ofLebesgue’s
measuereonR eareidentical.Thus,choosingfork theincreasingrearrangement
+
h , (resp.: the decreasing rearrangement h ) of h, we obtain that there exists
∗
∗
essentially a unique increasing function h , (resp. a unique decreasing function
∗
h ) such that:
∗
(law)
Γ(h) = Γ(h )=Γ(h ) (15)
∗
∗
Werecallthatthe functionh (resp.the functionh )istheunique(equivalence
e∗ e e
∗
class of) increasing (resp. decreasing) function such that for every a 0:
≥
meas x; h(x)<a =meas x, h∗(x)<a =meas x; h (x)<a
∗
where meas(cid:0)indicates Leb(cid:1)esgue’s m(cid:0)easure on R (cid:1). (cid:0) (cid:1)
+
From Proposition 1.1, Γ(h) is a GGC r.v. and we shall denote by µ the
h
Thorin measure associated with Γ(h).
e
e
1.3. m-Wiener-Gamma integrals, (m,G) GGC r.v.
In this work, we shall often consider a GGC r.v. whose associatedThorin mea-
sure has finite total mass. Thus, we shall now particularize Proposition 1.1 in
this case.
1.3.a Let m>0 and h:[0,m] R a Borel function such that:
+
→
m
log 1+h(u) du< (16)
∞
Z0
(cid:0) (cid:1)
We call m-Wiener integral of h the r.v.:
m
Γ (h):= h(u)dγ (17)
m u
Z0
Since (γ , 0 u m) and (eγ γ , 0 u m) have the same law, we
u m (m u)
deduce from≤(17),≤after making t−he ch−ang−e of≤varia≤ble m u=v:
−
m m
(law) (law)
Γ (h) = h(u)dγ = h(m u)dγ (18)
m u u
−
Z0 Z0
e
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 354
We note, in relation with (15) above, that if h is increasing, resp. decreasing,
then u h(m u) is decreasing, resp. increasing.
→ −
1.3.b Let m>0 and G be a positive r.v. such that:
E log+(1/G) < (19)
∞
(cid:0) (cid:1)
We say that a positive r.v. Γ is a (m,G) GGC if:
E(e−λΓ)=exp m ∞(1 e−λx)dx E(e−xG) (20)
− − x
n Z0 o
Of course, from (7), a (m,G) GGC r.v. is a GGC r.v. whose Thorin measure µ
equals:
µ(dx)=mP (dx) (21)
G
where P denotes the law of G and we have:
G
µ ]0, [ =m (22)
∞
(cid:0) (cid:1)
Under (21), it is clear that:
µ(dx)
logx µ(dx)< and <
| | ∞ x ∞
Z]0,1] Z[1,∞[
E log+(1/G) < (23)
⇔ ∞
(cid:0) (cid:1)
We denote by Γ (G) the (law of the) r.v. Γ defined by (20). Hence:
m
E[e−λΓm(G)] = exp m ∞(1 e−λx)dxE(e−xG) (24)
− − x
n Z0 o
λ
= exp mE log 1+ (25)
− G
n (cid:16) (cid:16) (cid:17)(cid:17)o
1.3.c G or 1/G ? How to choose ?
WehaveusedthenotationΓ (G)duetotherelation(24).Ofcourse,therelation
m
(25)invites,onthe contrary,toadoptthe notationΓ (1/G).However,weshall
m
notadoptthislatternotationasthenotationΓ (G)isusedbyL.Bondesson[5]
m
who has contributed in an essentialmanner to the study of the GGC variables.
1.3.d Proposition 1.1, when the Thorin measure has a finite total mass m,
admits the following translation.
Proposition1.3Ar.v.isa(m,G)GGCifandonlyifitisam-Wienerintegral.
More precisely:
(law)
1) If Γ (G) = Γ (h), then
m m
1
h(u)= e u [0,m] (26)
FG−1(u/m) ∈
(cid:0) (cid:1)
L.F. James, B. Roynette and M. Yor/GGC and Dirichlet Means 355
where FG−1 denotes the right continuous inverse, in the sense of composition of
functions, of F , the cumulative distribution function of G.
G
(law)
2) If Γ (h) = Γ (G), then
m m
(law) 1
G = (27)
h(U )
m
where U denotes a uniform r.v. on [0,m].
m
1.3.e Some classical results
We gather here some results which are due to L. Bondesson [5] and which we
shall use in the sequel. Let m > 0 and G satisfy (19). Then, denoting f
Γm(G)
the density of Γ (G):
m
xm 1
−
f (x)= g(x) (x>0) (28)
• Γm(G) Γ(m)
where g is a completely monotone function ([5], p. 49). Moreover ([5], p. 50) g
admits a limit on the right of 0 and:
g(0 )=exp mE(logG) (29)
+
We note that, since by hypothesis E (cid:8)log+(1/G) <(cid:9) , g(0 ) is finite if and
+
∞
only if E log+(G) < . In Section 2 of this work (see Theorem 2.1) we give
∞ (cid:0) (cid:1)
an explicit form of g when E logG < .
(cid:0) (cid:1) | | ∞
m may be determined from the knowledge of f :
• (cid:0) (cid:1) Γm(G)
f (x)
m=sup α 0; lim Γm(G) =0 (30)
n ≥ x↓0+ xα−1 o
(see [5], p. 51).
1.4. m-Wiener Gamma integrals, m Dirichlet means, Gamma (m)
mixtures
1.4.a The preceding discussion leads us to introduce, for every m > 0, the
Dirichlet process with parameter m, denoted: (D(m), 0 u m) and defined
u
≤ ≤
as:
γ
(D(m), 0 u m)= u, 0 u m (31)
u ≤ ≤ γ ≤ ≤
m
(cid:16) (cid:17)
It is well known, and it is an easy consequence of the properties of the “beta-
gamma algebra” that this process (D(m), 0 u m) is independent from the
u
≤ ≤
r.v. γ , hence from (γ , v m). Indeed, if γ and γ are two independent
m v a b
≥
gamma variables with parameter a, resp. b, the basic “beta-gamma algebra”
states that:
γa (law)
, γ +γ = β ,γ
a b a,b a+b
γ +γ
(cid:18) a b (cid:19)
(cid:0) (cid:1)