What Are Cumulants ?
Dediated to Professor DietrihMorgenstern,
onthe oasion of hisseventy-fifth birthday
LutzMattner
Reeived: July5,1999
CommuniatedbyFriedrihGotze
Abstrat. LetP bethesetofallprobabilitymeasuresonRpossess-
ingmomentsofeveryorder. ConsiderP asasemigroupwithrespet
to onvolution. AftertopologizingP in anaturalway,wedetermine
all ontinuous homomorphisms of P into the unit irle and, as a
orollary, those into thereal line. The latterare preisely thenite
linear ombinationsof umulants, and from these allthe former are
obtainedviamultipliationbyiandexponentiation.
We obtain as orollaries similar results for the probability mea-
sureswith someor nomomentsnite, andharaterizationsof on-
stantmultiples ofumulantsasaÆnelyequivariantandonvolution-
additive funtionals. The \no moments"-ase yields a theorem of
Halasz. Otherwiseourresultsappeartobenewevenwhenspeialized
toyieldharaterizationsoftheexpetationorthevariane.
Ourbasitoolisarenementoftheonvolutionquotientrepresenta-
tiontheoremforsignedmeasuresofRuzsa&Szekely.
1991MathematisSubjetClassiation: 60E05,60E10,60-03.
Keywordsand Phrases: Additive funtional, harateristi funtion,
harater, onvolution, equivariane, expetation, Halasz, histori-
al note, homomorphism, mean, moment, multipliative funtional,
Ruzsa,semi-invariant,semiinvariant,semigroup,Szekely,variane.
1 Introdution,results,andeasy proofs
1.1 Aim. Cumulants are ertain funtionals of probability measures. This
paperattemptstoexplainmorepreiselywhattheyarebyharaterizingthem
throughtheirmostusefulproperties. Forsimpliity,onlytheone-dimensional
ase of probability measures onR is treated. There themost familiar exam-
ples of umulantsare theexpetation and the variane. Our resultsyield, in
partiular,newdesriptionsof therolesplayedbytheselattertwofuntionals
in probabilitytheory.
1.2 Guide. Thedenitionofumulantsisrealledin Subsetion1.4below,
asformula(4). Theusefulpropertiesofumulants,referredtoabove,are the
homomorphism property(5) and their transformationbehaviourunder aÆne
mappings, (14). The relation betweenumulants and momentsis realled in
Subsetion1.5.
Subsetion1.6introduestopologiesonthedomainsofdenitionoftheumu-
lants,withtheaimofformulatingregularityassumptionsinourtheoremsand
orollaries. That someregularityassumptions areatually neessary, atleast
in theresults1.8{1.12,isdemonstratedin 1.20.
Theorem 1.8, haraterizing the ontinuous haraters of the semigroup
Prob
1
(R) , is the main result of the present paper. Its natural forerunner
fromtheliterature,namelythetheoremofHalasz,isrealledin1.10belowas
aspeialaseofCorollary1.9.
Another orollary of Theorem 1.8, and perhaps the most interesting result
of this paper, is the haraterization of the nite linear ombinations of u-
mulants asthe ontinuous, R-valued, and onvolution-additivefuntionals of
probabilitymeasures,statedinTheorem1.11andCorollary1.12. Suhresults
wereonjetured byKemperman(1972). Byrestritingthe funtionalsto be
[0;1[-valued,wearriveataharaterizationofthevarianein1.14. [Arelated
resultofMartinDiaz(1977)isdisussedin 1.22.℄
Our next results, 1.17 and 1.18, are spezializations of 1.8 and 1.11 to sale
equivariantfuntionals,thedenitionofwhihbeingrealledin1.16.
Asafurtherorollary,weobtainin1.19aharaterizationoftheexpetationas
theonlynontrivialontinuousfuntionalhomomorphiwithrespettoadditive
andmultipliativeonvolutions.
HistorialandetymologialremarksonumulantsaregiveninSubsetion1.21.
Subsetion1.22disussessomefurtherreferenesrelatedto thepresentwork.
Easy proofs are given immediately after the statement of a result in Setion
1. Theonly diÆultproofof thispaper,needed forthe\only if"partof our
main result1.8, is the ontent of Setion 2. Its basi tehnial tool, rening
theonvolutionquotientrepresentationtheoremforsignedmeasuresofRuzsa
&Szekely(1983,1985,1988),issuppliedin Subsetion2.5.
1.3 Some notation andonventions. Thepositiveintegersaredenoted
byN, thenonnegativeonesbyN
0 .
If X is a set equipped with a -algebra A, we let Prob(X) denote the set
of all probability measures dened on A. The real line R is understood to
be equippedwith itsBorel -algebra. The onvolutionof P;Q2 Prob(R) is
denoted byPQ. Wewrite Æ
a
fortheDirameasure onentratedat a2R,
and Æ := Æ
0
for the one onentrated at zero. For the image measure of a
probability measure P under a measurable funtion f, we use the notation
fP. We write suppP for the support [= minimal losed set of probability
one℄ofaP 2Prob(R) .
Prob(R) willmainlybeonsideredasasemigroupwithrespettoonvolution.
Homomorphisms of a semigroup [below always a sub-semigroup of Prob(R) ℄
into themultipliativegroupTof omplexnumbersofabsolutevalueonewill
bealledharaters,homomorphismsinto theadditivegroupR will bealled
additive funtions.
1.4 Cumulants. Wepresentbelowtheusualintrodutionofumulantsand
their most important properties. For P 2 Prob(R) , let b
P denote its Fourier
transformorharateristifuntion,denedby
b
P(t) :=
Z
e itx
dP(x) (t2R): (1)
The most importantreason for onsidering Fouriertransforms of probability
measuresismultipliativitywithrespettoonvolution:
(P Q)b(t) = b
P(t) b
Q(t) (P;Q2Prob(R); t2R) : (2)
Letlog denotetheusuallogarithmdened on,say,fz2C : jz 1j<1g. Let
P 2Prob(R). Then b
P is ontinuouswith b
P(0)=1,sothat logÆ b
P is dened
in someP-dependentneighbourhoodofzero. Nowput
Prob
r
(R) :=
P 2Prob(R) : Z
jxj r
dP(x)<1
(r2N
0 ); (3)
andassumethatr2N andP 2Prob
r
(R). Then b
P andthuslogÆ b
P isr times
ontinuouslydierentiableintheneighbourhoodofzerointroduedabove,and
thenumber
r
(P) := i r
D r
logÆ b
P
(0) (4)
isalled therthumulant ofP. [Readerswonderingaboutthisstrangename
arereferredtoSubsetion1.21.℄ Itiseasytoshowthattheumulantsarereal-
valued funtionals. Their most important property, whih obviously follows
from(2) and(4),isadditivitywith respettoonvolution:
r
(P Q) =
r
(P)+
r
(Q) (r2N; P;Q2Prob
r
(R)) : (5)
Inotherwords: Foreahr2N, (Prob
r
(R );) isasemigrouponwhih
r isan
additivefuntion.
1.5 Examples,expressionintermsofmoments. Thetwomostfamiliar
examplesofumulantsarethemeanandthevariane 2
,sine
1
(P) = (P) :=
Z
xdP(x) (P 2Prob
1 (R ));
2
(P) = 2
(P) :=
Z
(x (P)) 2
dP(x) (P 2Prob
2 (R) ):
These formulas are speial ases of the relation between umulants and the
moments
r (P) :=
R
x r
dP(x) = i r
(D n
b
P)(0) (r 2N
0
;P 2Prob
r (R )):
Onepossibilityofexpressingthisrelationistousethereursion
r+1
= r
X
l=0
r
l
r l
l+1
(r2N
0
); (6)
whihiseasilyprovedusingtheLeibnizruleforthedierentiationofaprodut
andtherepresentationofthemomentsasderivatives: ForP 2Prob
r+1 (R ) put
' :=
b
P and := log', in a neighbourhood of zero, and ompute D r+1
' =
D r
('D )= P
r
l=0 r
l
(D r l
')(D l+1
),evaluatetheextremeleftandright
hand sides atzero, anddivide byi r+1
, to arriveat (6). Sinethe oeÆients
of
r+1 and
r+1
in (6)arebothone,itfollowsbyindution that
r
=
r
+ polynomialwithoutonstanttermin
1
;:::;
r 1
(r2N); (7)
andthatorrespondingrelationsholdwhenandareinterhanged. Various
expliitfomulasderivedfromtheserelationsandsomeexamplesofatualom-
putationsof umulantsmay be foundin Chapter 3ofKendall,Stuart &Ord
(1987). Wemerelynote heretwofurther examples,foronveniene rewritten
in termsofenteredmoments,
3
(P) = Z
(x (P)) 3
dP(x) (P 2Prob
3 (R ));
4
(P) = Z
(x (P)) 4
dP(x) 3 2
(P)
2
(P 2Prob
4 (R )):
As one might suspet on seeing these formulas, the variane
2
is the only
nonnegativeumulant. [Thisfatfollowseasilyfrom1.13below,asanbeseen
fromtheproofof1.14.℄
1.6 Topologies on some subsets of Prob(R). One of our aims is to
show that every \reasonable"homomorphism from (Prob
r
(R);) into (R;+)
is a linearombination of umulants of order at most r. This is theontent
of Corollary 1.12, where \reasonable"is speied to mean \ontinuous". To
this end we introdue here on eah Prob
r
(R ) a topology. In order to make
theontinuityassumptionin Corollary1.12weak,wehavetohooseastrong
topologyonProb
r
(R). Wetaketheoneinduedbytheweightedtotalvariation
metrid
r
dened by
d
r
(P;Q) :=
Z
(1+jxj r
)djP Qj(x) (P;Q2Prob
r
(R )): (8)
Wefurtheronsider
Prob
1
(R) :=
\
r2N
0 Prob
r (R);
whih is the largest set of probability measures on whih every umulant is
dened. WetopologizeProb
1
(R) bythefamilyofmetris(d
r :r2N
0 ).
1.7 Lemma. a)Eah
r jProb
r
(R) isontinuouswith respettod
r .
b) Letr 2N and 2℄0;1[. Then there existsasequene (P
n
) in Prob
1 (R )
with
lim
n!1 d
r 1 (P
n
;Æ) = 0; (9)
lim
n!1
l (P
n
) = 0 (l=1;:::;r 1); (10)
lim
n!1
r (P
n
) = : (11)
Proof. a) The funtionals (
l
:l =1;:::;r) are obviously ontinuouswith
respet to d
r
, and(7) showsin partiular that
r
is apolynomialfuntion of
them.
b) We may restrit attentionto those n 2 N with n r
1 and put P
n :=
(1 n r
)Æ +n r
Æ
n
. Then P
n
2 Prob
1
(R) , and d
r 1 (P
n
;Æ) =
n yields
(9). Bypart a), (9) implies (10). Finally, (11) follows from
l (P
n ) =n
l r
(l=1;:::;r) and(7).
1.8 Theorem (ontinuous haraters of Prob
1
(R )). A funtion
jProb
1
(R) isaontinuousharateri
(P)=exp(i X
l2N
l
l
(P)) (P 2Prob
1
(R)) (12)
holds for some nitely supported sequene of real numbers (
l
: l 2 N). The
latter, ifexistent, isuniquelydeterminedby .
Proof. Theproofof the\onlyif"partistheontentofSetion 2. The\if"
partfollowstriviallyfrom 1.7a)and(5).
Finally suppose that we have (12) and an analogous representation of in-
volvinganother nitely supported sequene (~
l
: l2 N). Then thesequene
(d
l ):=(
l
~
l
)yieldsananalogousrepresentationoftheonstantharater1.
Supposethatnotalld
l
vanish. Putr:=minfl:d
l
6=0gandapply1.7b)with
:= =jd
r
j. Then 1 = exp(i P
r
l=1 d
l
l (P
n
)) ! exp(i) = 1for n ! 1.
This ontraditionshowsthat wemusthaved
l
=0foreveryl2N, aswasto
beproved.
1.9 Corollary. Let r2N
0
. A funtion jProb
r
(R ) is aontinuous har-
ater i (12) holds with
l
= 0 for l > r, and with Prob
r
(R) in plae of
Prob
1 (R) .
Proof. Again,the\if"partfollowsfromfrom1.7a)and(5). Toprove\only
if": Let jProb
r
(R) be a ontinuous harater. Then, by 1.8, its restrition
jProb
1
(R) fulls (12) for some nitely supported sequene (
l
). Assume
that
l
6= 0 for some l > r. Put ~r := minfl2N :
l
6=0g. Choose (P
n )
aording to 1.7 b) with ~r in plae of r and with := =j
~ r
j. Then, sine
r < r,~ we have P
n
! Æ with respet to d
r
. On the other hand, we have
(P
n
)! 16=1=(Æ). This ontraditiontotheontinuityofshowsthat
we must have
l
= 0 for l > r. It follows that the right hand side of (12)
isdened andontinuousonProb
r
(R) . SineProb
1
(R ) isobviouslydensein
Prob
r
(R),thisimpliesthat(12)alsoholdswithProb
r
(R)inplaeofProb
1 (R) .
1.10 Theoremof Hal
asz. Thelastorollaryyieldsinpartiularatheorem
ofHalasz,presentedonpage132ofRuzsa&Szekely(1988),whihreads:
1isthe onlyharaterof Prob(R) ontinuouswith respetto weak
onvergene.
Infat,thespeialaser=0ofourCorollary1.9isslightlystronger,sineour
ontinuityassumptionreferstoastrongertopologyonProb(R).
1.11 Theorem (additive funtions on Prob
1
(R)). A funtion
jProb
1
(R) !R isontinuousand additive i
(P)= X
l2N
l
l
(P) (P 2Prob
1
(R)) (13)
holdsforsomenitelysupportedfamilyofrealnumbers(
l
:l2N). Thelatter,
if existent,isuniquely determined by.
Proof. The\if"partandtheuniquenessof(
l
)followsviamultipliationby
iandsubsequentexponentiationfromtheorrespondingstatementsin1.8.
Toprovethe\onlyif"part,letjProb
1
(R) !R beontinuousandadditive.
Put
(P) := exp(i(P)) (P 2Prob
1 (R)):
Then satisesthehypothesisof Theorem1.8,and heneanberepresented
asin(12). Thisimplies
(P) = (P)+ X
l
l
l
(P) (P 2Prob
1 (R)) ;
where jProb
1
(R) ! 2Z. Sine must be additive, (Æ) = 0. Sine
mustbeontinuousandProb
1
(R) isonvex,(Prob
1
(R))mustbeonneted.
[Here we have used the obviousfat that for P;Q 2 Prob
1
(R ) the funtion
[0;1℄3t7!tP+(1 t)Q2Prob
1
(R) isontinuous.℄ Thus =0.
1.12 Corollary. Let r 2 N
0
. A funtion jProb
r
(R ) !R is ontinuous
andadditive i(13)holdswith
l
=0for l>r andwith Prob
r
(R) in plaeof
Prob
1 (R) .
Proof. Dedue1.12from1.9,byarguingasintheproofof1.11. Alternatively,
dedue1.12from1.11byarguingasintheproofof1.9.
1.13 Lemma (umulantsof Bernoullidistributions). Forr2N, let
f
r
j[0;1℄!R bedenedby
f
r
(p) :=
r
((1 p)Æ
0 +pÆ
1
) (p2[0;1℄):
Then,for eahr,f
r
isapolynomial funtionofdegreerwithrsimplezerosin
[0;1℄.
Proof. Itisknown[forexample,fromKendall,Stuart&Ord(1987),exerise
5.1℄that
f
r+1
(p) = p(1 p)f 0
r
(p) (r2N; p2[0;1℄);
where theprimedenotes dierentiationwith respetto p. Sinef
1
(p)=pfor
p2 [0;1℄, thelaim follows byan indution argument, using Rolle'stheorem
andthefatthat f 0
r
hasat mostr 1zeros,ountingmultipliity.
1.14 A haraterization of the variane. Afuntion jProb
1 (R) !
[0;1[isontinuousandadditive i=
2
for some2[0;1[.
Proof. The \if" laim is trivial. To prove \only if", we may by Theorem
1.11startfromtherepresentation(13). InsertingthereP =Æ
a
witha2R,we
see thattheassumption 0fores
1
=0. Thus, exeptforthetrivialase
=0,wehave
(P)= r
X
l=2
l
l
(P) (P 2Prob
1 (R))
for some r 2 with
r
6= 0. Suppose now that r 3. Then we may, by
the lemma 1.13, hoose a Bernoulli distribution P
0
= (1 p)Æ
0 +pÆ
1 with
r
r (P
0
) < 0. It followsthat (P) < 0for P := (x 7! ax) P
0
with a > 0
suÆientlylarge,using(14)below. Thisontraditionprovesourlaim.
1.15 Affine equivariane of umulants. The seond most important
propertyoftheumulantsistheirbehaviourunderaÆnetransformations: For
r2N, P2Prob
r
(R) anda;b2R, wehave
r
((x7!ax+b)P) =
a
1
(P)+b (r=1);
a r
r
(P) (r2):
(14)
Inpartiular,eahumulantisaÆnelyequivariantinthesenseofthefollowing
denition and,byatrivialspeialization,alsosaleequivariant.
1.16 Definition (equivariane). a) Let X be a set and T be a set of
funtionsfromX intoX. Afuntion'jX isalledequivariant,withrespetto
T,ifwehavetheimpliation
x;y2X;'(x)='(y);T 2T =) '(T(x))='(T(y)): (15)
b)Fora;b2R deneT
a;b
jProb(R) !Prob(R) by
T
a;b
(P) := (x7!ax+b)P (P 2Prob(R))
and put T :=fT
a;b
:a;b2Rg. LetP Prob(R) satisfy theimpliation P 2
P;T 2T =) T(P)2P. Thenafuntion'jP isalledaÆnelyequivariantif
itisequivariantwithrespettoT,in thesenseofparta).
) Wedene afuntion 'jP tobesale equivariantifitsatisesthedenition
givenin b)above,but withb=0anda>0inthedenitionof T.
1.17 Theorem (equivariant ontinuous haraters of Prob
1 (R)).
A funtionjProb
1
(R ) isasaleequivariantontinuousharateri
(P) = exp(i
r
(P)) (P 2Prob
1
(R)) (16)
for somer2N andsome2R.
Proof. The \if" partis trivial. To prove\only if": Dene S
a
(P) :=(x 7!
ax) P for P 2 Prob(R) and a 2℄0;1[. For 2℄0;1[, let P
denote the
Poissondistributionwithexpetation. Then
l (S
a (P
)) = a l
(l2N; a;2℄0;1[): (17)
Now let jProb
1
(R) be a sale equivariant ontinuous harater. Applying
1.8, weget(12)forsomenitelysupported sequene(
l
:l2N) , andwehave
to showthatthereisat mostonel2N with
l
6=0. Using (17),(12)yieldsin
partiular
(S
a (P
)) = exp(ip(a)) (a;2℄0;1[) (18)
wherepisthepolynomialfuntiondened by
p(a) :=
X
l2N
l a
l
(a2C):
Now assume, to get a ontradition, that there are at least two l 2 N with
l
6=0. Thenforarbitrarya
1
;a
2
2℄0;1[witha
1 6=a
2
andarbitrary
1
;
2 6=0,
there existsanumberb2℄0;1[with
1 p(ba
1
)
2 p(ba
2
) 2= 2Z: (19)
[Proof: Assume withoutlossof generalitythat a
1
<a
2
. If ourlaim is false,
thentherationalfuntionC 3z7!R (z):=p(a
1 z)=p(a
2
z)isonstant. Butby
ourassumptiononp,%:=supfjzj:z2C; p(z)=0g>0. Inviewof 0<a
1
<
a
2
, itis obvious that R hasazero, namelyon theirle fjzj=%=a
1
g. Hene
R0andthusp0,aontradition.℄
Nowhoosespeiallya
1
;a
2
2℄0;1[witha
1 6=a
2
insuhawaythatp(a
1 )
p(a
2
)>0. Choose
1
;
2
2℄0;1[with
1 p(a
1
) =
2 p(a
2
); (20)
hooseb asin (19),and putQ
k :=S
a
k (P
k
) fork=1;2. Then (18)and (20)
yield (Q
1
) = (Q
2
), whereas(18) also yields (S
b (Q
k
)) = (S
ba
k (P
k )) =
exp(i
k p(ba
k
)) for k = 1;2, so that (19)implies (S
b (Q
1
)) 6= (S
b (Q
2 )), in
ontraditiontothesaleequivarianeof.
1.18 Theorem(saleequivariantadditivefuntionsonProb
1 (R)).
A funtionjProb
1
(R) !R isontinuous, additive,andsale equivariant, i
thereexistr2N and2R suhthat=
r .
Proof. Proeedasintheproofofthe\onlyif"partofTheorem1.11,butuse
equivarianeofand1.17inplaeof1.8.
1.19 A haraterization of the expetation. Notation: Inthis sub-
setiononly,wewritePQfortheusualonvolutionPQofP;Q2Prob(R) ,
and P Q for the multipliative onvolution of P;Q 2 Prob(R ), that is, the
distributionofXY withX;Y beingindependentrandomvariableswithdis-
tributionsP;Q.
Theorem. Let jProb
1
(R )!R beontinuous. Then wehave both
(PQ) = (P)+(Q); (21)
(P Q) = (P)(Q) (22)
for P;Q2Prob
1
(R ), ieither=
1
or =0.
Proof. The\if"partisobvious. Soassumethatisontinuousandsatises
(21)and (22). Byapplying(22) to Q=Æ
a
, for everya2℄0;1[, wesee that
issale equivariant. Hene (21)and Corollary1.18yield =
r
for some
2R andsomer2N. ChooseP 2Prob
1
(R) with
r
(P)6=0,forexampleP
=Poissondistributionwith parameter1. InsertthisP and Q=Æ
1
into (22),
use =
r
, anddivide by
r
(P). Theresultis = 2
r (Æ
1
). If r2,then
r (Æ
1
)=0,hene=0andthus=0. Ifr=1,then
r (Æ
1
)=1,heneeither
again=0and=0,or=1andthus=
1 .
1.20 \Counterexamples". Examplesa) andb)belowshowthattheon-
tinuityassumptionsin1.8{1.12annotbeomittedwithoutsubstitute. Both
a)andb)shouldberegardedaspathologial. Ontheotherhand,theexamples
in)showthatnotonly1.8{1.12,butalso1.14and,using(23),also1.17and
1.18reeivenon-pathologialounterexamplesifthe ontinuity assumptionis
dropped and if the domain of denition of the funtionals is taken to be to
small. Conerning 1.8 {1.12, we mayalso refer to exampled), suggestedto
me byI.Z.Ruzsa,where thedomainof denition ofouldbethoughtofas
beingnotmuhsmallerthanProb
1 (R) .
a) By the axiom of hoie, there exists a disontinuous additive funtion
fjR ! R. Now (P) := f((P)) denes a disontinuous additive funtion
jProb
1
(R) !R.
b)[Ruzsa&Szekely(1988),pp.122-123,2.3and2.4℄onstrut,usingtheaxiom
ofhoie, ahomomorphismfrom(Prob(R );) into(R;+) whihextends the
expetation
1
dened onthe subsemigroup Prob
1
(R) . They also show that
eah suh assumes negative values for some P with support in [0;1[. It
followsthattheonstrutedisadisontinuousadditivefuntionfromProb(R )
into R.
) Onthesemigroup
Prob
(R) := fP 2Prob(R) : suppP ompatg Prob
1 (R)
weobtainanadditiveandnonnegativefuntional,normalizedhereastosatisfy
additionallyonditionii)from1.22below,byeahofthefollowingdenitions:
(P) :=
1
2
(maxsuppP minsuppP) (P 2Prob
(R )); (23)
(P) :=
log b
P(i)+log b
P( i)
2logosi
(P 2Prob
(R) ): (24)
[In(24),weuseofoursethedenition (1)withC in plaeofR.℄
d)Considerthesemigroup
P :=
n
P 2Prob
1 (R) :
b
P holomorphinearzero o
Prob
1 (R):
Let(a
l
:l2N)beanysequeneofrealnumberssatisfyinga
l
=O("
l
),forevery
">0. Then
(P) :=
1
X
l=1 a
l
l!
l
(P) (P 2P) (25)
denes anadditivefuntion onP. [Tosee that theseries in (25)alwayson-
verges, observe that logÆ b
P is now holomorphi in some P-dependent neigh-
bourhood of zero, sothat its Taylorseries P
1
l=1
l
(P)(iz) l
=l! onvergesfor
jzjsuÆientlysmall.℄
1.21 Some early history and etymology. Cumulantswereapparently
rstintroduedbyT.N.Thiele[1838-1910℄underthenameof\half-invariants".
Hald (1981)desribes, onpages7-10, Thiele'sontributionsand theirinsuÆ-
ient aknowledgement by K. Pearson and R.A. Fisher. Aording to Hald,
umulants arerst dened in thebook Thiele (1889). [This I did nothek.
Hald's formula (4.1), laimed to be Thiele's denition, is, up to an obvious
misprint,thenowwell-knownreursion(6),determining
r+1
asapolynomial
in themoments
l
.℄ Inalaterandmoreaessibleversionofhis book,Thiele
(1903)essentiallygivesdenition(4). Hald(1998)ontainsamuhmoreom-
prehensiveearlyhistoryofumulants.
Later authors, suh as Craig (1931) and Wishart (1929), refer to the umu-
lantsas\semi-invariantsofThiele",whileFisher(1929-30),onpage200ofhis
paper,simplyallsthem\semi-invariants",withoutbotheringtonameThiele.
But Wishart and Fisher, who obviously new about eah others work before
publiation, prefer to use the new term \umulative moment funtions" in-
stead. The reason for adopting this term is hinted at in Fisher'spaper: On
page 199, he gives an interesting although perhaps not quite preise deni-
tion of rather general \moment funtions" of populations, roughly speaking
bypolynomialestimability,whih seemsat anyratetobeintendedtoinlude
polynomialfuntionsofnitely manyordinarymoments,andheneinparti-
ular umulants. On page 202, Fisherthen refers to a \umulativeproperty"
of the logarithm of the Laplae transform whih, expressed in terms of the
umulants, is just ondition(5). Thus the theadjetive \umulative" refers,
in this ontext, to ahomomorphism ondition. In partiular, it is not used
to distinguishaoneptrelatedtoprobabilitymeasuresfrom aorresponding
oneptrelatedtoprobabilitydensities,aswouldoftenbetheaseintheolder
statistialliterature.
Finally, \umulative moment funtion" was abbreviated to "umulant" by
Fisher & Wishart (1931-32) and Fisher (1932), with Hotelling (1933) laim-
ingtohavesuggestedthisname,whihquiklybeamethestandardoneinthe
englishlanguageliterature. Therstpubliationhavingtheword\umulant"
in itstitle seemstobethepaperbyCornish&Fisher(1937),whorepeatthe
denition, but already Haldane (1937), page 136, uses \umulants" without
denition orreferene.
Readersgenerallyinterestedinthehistoryofprobabilistiorstatistial terms
arereferredto David (1995,1998)asausefulstartingpoint.
1.22 Related work not disussed above. The following papers have
somerelationwiththepresentone.
Craig(1931)statesonpage160aforerunnerofourCorollary1.18. Wherewe
assumemereontinuityof,Craigassumesinpartiularthatisapolynomial
funtion of some nite number of moments
l
. His treatment is not quite
rigorous: Forexample,nodomainofdenitionofisspeied,hisonlusion
is=
r
forsomer[insteadoftheorretonlusion=
r
forsomerand
℄,andaproofisoeredonlyfor theasewhereisapolynomialfuntionof
1
;:::;
4 .
Savage(1971)haraterizesmomentsand moregeneralexpetations of expo-
nentialpolynomialsasfuntionalssatisfying,ontheonehand,onditionslike
(P Q)=T((P);(Q))withT unspeiedand, ontheotherhand,having
a representation (P) = R
fdP with f unspeied. His rst assumption is
moreliberalthanourhomomorphismassumptions,buthisseondassumption
isratherrestritive,exludingforexampleeveryumulant
r
withr2. Thus
theworkof Savageisinomparabletothepresentone.
Martin Diaz (1977), Teorema 4, states a haraterization of the vari-
ane whih may be formulated as follows. We temporarily put P :=
fP 2Prob(R) : suppP nite g.
Theorem (Martin Diaz)LetjP