Functional Equations and Characterization Problems on Locally Compact Abelian Groups

269 Pages·2008·1.74 MB·English

by  Gennadiy Feldman

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Gennadiy Feldman
Functional Equations
and Characterization
Problems on Locally
Compact Abelian Groups
S E
M M
E S
SS EE E M S
MM MM uropean athematical ociety

EE SS
feldman_titelei 31.3.2008 17:10 Uhr Seite 4
Author:
Gennadiy Feldman
Mathematical Division
B. Verkin Institute for Low Temperature Physics and
Engineering of the National Academy of Sciences of Ukraine
47 Lenin Ave.
Kharkov, 61103, Ukraine
E-mail: [email protected]
2000 Mathematics Subject Classification: 60-02, 60B15, 62E10; 43A05,
43A25, 43A35, 39B52
Key words: Locally compact Abelian group, Gaussian distribution,
characterization problems of mathematical statistics, functional equation
ISBN 978-3-03719-045-6
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9 8 7 6 5 4 3 2 1
Preface
Characterization problems in mathematical statistics are statements in which the de-
scriptionofpossibledistributionsofrandomvariablesfollowsfrompropertiesofsome
functionsinthesevariables. Oneofthefamousexamplesofacharacterizationprob-
lem is the classical Kac–Bernstein theorem ([65], [13]). This theorem characterizes
a Gaussian distribution by the independence of the sum (cid:2) C (cid:2) and of the differ-
1 2
ence (cid:2) (cid:2)(cid:2) of independent random variables (cid:2) . Taking into account that the char-
1 2 j
acteristic function of the random variable (cid:2) with distribution (cid:3) is the expectation
j j
f .y/ D (cid:3)O .y/ D EŒei(cid:2)jy(cid:4), it is easily verified that the Kac–Bernstein theorem is
j j
equivalenttothestatementthat,intheclassofnormalizedcontinuouspositivedefinite
functions,allsolutionstotheKac–Bernsteinfunctionalequation
f .uCv/f .u(cid:2)v/Df .u/f .v/f .u/f .(cid:2)v/; u;v 2R;
1 2 1 1 2 2
areoftheformf .y/Dexpf(cid:2)(cid:5)y2Cib yg,where(cid:5) (cid:3)0,andb 2R.
j j j
TheKac–Bernsteintheoremwasthefirstamongcharacterizationtheoremswhere
independent linear forms of independent random variables (cid:2) under different restric-
j
tionson(cid:2) werestudied. ThesestudieswerecompletedwiththefollowingSkitovich–
j
Darmois theorem ([98], [23]): Let (cid:2) , j D 1;2;:::;n, n (cid:3) 2, be independent
j
random variables, and ˛ , ˇ be nonzero real numbers. If the linear forms L D
j j 1
˛ (cid:2) C(cid:4)(cid:4)(cid:4)C˛ (cid:2) and L D ˇ (cid:2) C(cid:4)(cid:4)(cid:4)Cˇ (cid:2) are independent, then all random
1 1 n n 2 1 1 n n
variables (cid:2) are Gaussian. Just as in the case of the Kac–Bernstein theorem, the
j
Skitovich–Darmois theorem is equivalent to the statement that, in the class of nor-
malizedcontinuouspositivedefinitefunctions,allsolutionstotheSkitovich–Darmois
functionalequation
Yn Yn Yn
f .˛ uCˇ v/D f .˛ u/ f .ˇ v/; u;v 2R;
j j j j j j j
jD1 jD1 jD1
areoftheform
f .y/Dexpf(cid:2)(cid:5) y2Cib yg; .1/
j j j
where(cid:5) (cid:3)0andb 2R.
j j
TheSkitovich–DarmoistheoremwasgeneralizedbyGhuryeandOlkin([54])tothe
multivariablecasewhen,insteadofrandomvariables,randomvectors(cid:2) inthespace
j
Rm are considered, and coefficients of the linear forms L and L are nonsingular
1 2
matrices. InthiscasetheindependenceofL andL alsoimpliesthatallindependent
1 2
randomvectors(cid:2) areGaussian. TheproofoftheGhurye–Olkintheoremisalsoreduced
j
tosolvingthecorrespondingfunctionalequation. Itshouldbenotedthatnonsingular
matricesaretopologicalautomorphismsofthegroupRm.
Next Heyde proved the following result ([61]): Let (cid:2) , j D 1;2;:::;n, n(cid:3)2,
j
be independent random variables, and let ˛ , ˇ be nonzero real numbers such that
j j
vi Preface
ˇ ˛(cid:2)1 ˙ˇ ˛(cid:2)1 ¤ 0 for all i ¤ j. If the conditional distribution of L D ˇ (cid:2) C
i i j j 2 1 1
(cid:4)(cid:4)(cid:4)Cˇ (cid:2) givenL D˛ (cid:2) C(cid:4)(cid:4)(cid:4)C˛ (cid:2) issymmetric,thenall(cid:2) areGaussian. This
n n 1 1 1 n n j
statement is closely related to the Skitovich–Darmois theorem. The Heyde theorem
isalsoequivalenttotheassertionthat, intheclassofnormalizedcontinuouspositive
definitefunctions,allsolutionstotheHeydefunctionalequation
Yn Yn
f .˛ uCˇ v/D f .˛ u(cid:2)ˇ v/; u;v 2R;
j j j j j j
jD1 jD1
areoftheform(1).
Inthelast30yearsmuchattentionhasbeendevotedtogeneralizingoftheclassical
characterization theorems into various algebraic structures such as locally compact
Abelian groups, Lie groups, quantum groups, symmetric spaces (see e.g., [2], [29],
[31]–[44], [46], [47], [49], [52], [55], [56], [64], [78]–[82], [84]–[87], [91], [93],
[94]).
These investigations were motivated, first of all, by the desire to find the natural
limits for possible extensions of the classical results. The present book is devoted
togeneralizationoftheKac–Bernstein, Skitovich–Darmois, andHeydecharacteriza-
tiontheoremstothecasewhereindependentrandomvariablestakevaluesinasecond
countablelocallycompactAbeliangroupX,andcoefficientsoflinearformsaretopo-
logicalautomorphismsofX. Itturnsoutthatthepossibilitytoproveacharacterization
theoremforX notonlydependsonthestructureofX butalsodeterminesitsstructure.
For example, assume that independent random variables (cid:2) and (cid:2) take values in X
1 2
andtheircharacteristicfunctionsdonotvanish,thentheindependenceof(cid:2) C(cid:2) and
1 2
(cid:2) (cid:2)(cid:2) impliesthat(cid:2) areGaussianifandonlyifX containsnosubgrouptopologi-
1 2 j
cally isomorphic to the circle group T. Note that in the case of groups as well as in
the classical case, the proof of this theorem can be reduced to solving of some func-
tionalequationintheclassofnormalizedcontinuouspositivedefinitefunctionsonthe
charactergroupofX.
Wedescribeandcommentonthemaincontentsofthebook.
ChapterIcontainsmainlywell-knownfactsfromabstractharmonicanalysisandthe
theoryofinfiniteAbeliangroups. InSection1wegivethebasicdefinitionsandconsider
some examples of locally compact Abelian groups. In particular, we describe all
subgroupsoftheadditivegroupoftherationalnumbersQ,thegroupsofp-adicintegers
(cid:6)p,anda-adicsolenoids†a. Weformulatestructuretheoremsforvariousclassesof
locallycompactAbeliangroupsanddescribethetopologicalautomorphismgroupsof
thegroupsRn,Tn,(cid:6)p,†a. ThebasicresultsofUlmtheoryforcountablep-primary
Abeliangroupsarealsopresented. InSection2wediscusssomeaspectsofprobability
distributions on locally compactAbelian groups (the Bochner theorem, properties of
characteristicfunctions,theLévy–Khinchinformula,idempotentdistributions).
ChapterIIisdevotedtoGaussiandistributionsonalocallycompactAbeliangroupX.
InSection3wedefineGaussiandistributionsandstudytheirproperties. InSection4
we describe locally compact Abelian groups where every Gaussian distribution has
onlyGaussianfactors(thegroupanalogueoftheCramértheoremofdecompositionof
Preface vii
aGaussiandistribution). InSection5westudypropertiesofcontinuouspolynomials
onlocallycompactAbeliangroups. WedescribelocallycompactAbeliangroupswhere
anydistribution(cid:3)withacharacteristicfunctionoftheform(cid:3)O.y/DeP.y/,whereP.y/
isacontinuouspolynomial,isGaussian(thegroupanalogueoftheMarcinkiewicztheo-
rem). ThegroupanaloguesoftheCramérandMarcinkiewicztheoremsarebasictools
for proving characterization theorems in Chapters III–VI. In Section 6 we consider
GaussiandistributionsinthesenseofUrbanik,i.e.,suchdistributionsonX whichany
character transforms into Gaussian distributions on the circle group T. We describe
locallycompactAbeliangroupsforwhichtheclassofGaussiandistributionscoincides
withtheclassofGaussiandistributionsinthesenseofUrbanik.
In Chapter III we study distributions of independent random variables (cid:2) and (cid:2)
1 2
taking values in a locally compact Abelian group X such that (cid:2) C(cid:2) and (cid:2) (cid:2)(cid:2)
1 2 1 2
are independent. In Section 7 we describe all groups X where such distributions
are invariant with respect to a compact subgroup K of X and that, under the natural
homomorphismX 7! X=K,induceGaussiandistributionsonthefactorgroupX=K.
This is the widest subclass of locally compact Abelian groups on which the Kac–
Bernsteintypetheoremcanbeproved. ItconsistsofallgroupsX havingtheconnected
componentofzerowithoutelementsoforder2.
If the connected component of zero of a group X contains elements of order 2,
then for such groups the following natural problem arises: to describe all possible
distributionsofindependentrandomvariables(cid:2) takingvaluesinX andsuchthatthe
j
sum(cid:2) C(cid:2) andthedifference(cid:2) (cid:2)(cid:2) areindependent. Wepresentasolutiontothis
1 2 1 2
probleminSection8forthegroupR(cid:5)T andthea-adicsolenoids†a. InSection9
westudydistributionsofindependentidenticallydistributedrandomvariables(cid:2) and
1
(cid:2) takingvaluesinalocallycompactAbeliangroupX suchthat(cid:2) C(cid:2) and(cid:2) (cid:2)(cid:2)
2 1 2 1 2
areindependent(GaussiandistributionsinthesenseofBernstein).
ChaptersIVandVaredevotedtosomegroupanaloguesoftheSkitovich–Darmois
theorem. Let (cid:2) , j D 1;2;:::;n, n (cid:3) 2, be independent random variables taking
j
valuesinalocallycompactAbeliangroupX,and˛ ,ˇ betopologicalautomorphisms
j j
ofX. PutL D˛ (cid:2) C(cid:4)(cid:4)(cid:4)C˛ (cid:2) andL Dˇ (cid:2) C(cid:4)(cid:4)(cid:4)Cˇ (cid:2) .
1 1 1 n n 2 1 1 n n
InChapterIVweassumethatthecharacteristicfunctionsofindependentrandom
variables (cid:2) do not vanish. We prove in Section 10 that independence of the linear
j
formsL andL impliesthatall(cid:2) areGaussianifandonlyifX containsnosubgroup
1 2 j
topologicallyisomorphictothecirclegroupT. Undertheconditionthatthecharac-
teristic functions of (cid:2) do not vanish, this is the widest subclass of locally compact
j
AbeliangroupsonwhichtheSkitovich–Darmoistheoremcanbeextended.
AssumethatagroupX containsasubgrouptopologicallyisomorphictothecircle
groupT. Thenthefollowingnaturalproblemarises: todescribeallpossibledistribu-
tionsofindependentrandomvariables(cid:2) takingvaluesinX andhavingtheproperty
j
that the linear forms L and L are independent. The remainder of Chapter IV is
1 2
devotedtosolutionofthisproblem. Itturnsoutthatifthecharacteristicfunctionsof
independent random variables (cid:2) with distributions (cid:3) do not vanish and L and L
j j 1 2
are independent, then the distributions (cid:3) can be replaced by their shifts (cid:3)0 in such
j j
viii Preface
a way that all (cid:3)0 are supported in the connected component of zero of the group X.
j
Hence the problem can be reduced to the case when X is connected. An important
characteristicofaconnectedlocallycompactAbeliangroupX isitsdimensiondimX.
IfdimX D1,thenX istopologicallyisomorphictooneofthefollowinggroups: the
real line R, an a-adic solenoid †a or the circle group T. If dimX D 2, then either
X containsnosubgrouptopologicallyisomorphictothecirclegroupT orX istopo-
logicallyisomorphictooneofthefollowinggroups: T2,R(cid:5)T or†a(cid:5)T. Assume
thatthenumberofrandomvariables(cid:2) isequalto2. InSection11wedescribeforthe
j
two-dimensionaltorusT2 allpossibledistributionsoftherandomvariables(cid:2) inthe
j
casewhenlinearformsL andL areindependent. Generallyspeaking,thesedistri-
1 2
butionsarenotGaussian,andwedescribe,inparticular,alltopologicalautomorphisms
˛ ,ˇ ofthetwo-dimensionaltorusT2 forwhichthecorrespondingdistributionsare
j j
Gaussian. InSection12wesolvethesameproblemforthegroupsR(cid:5)T and†a(cid:5)T.
InChapterVweomittheassumptionthatthecharacteristicfunctionsofindependent
randomvariables(cid:2) donotvanishandsupposethat(cid:2) takevaluesindifferentclasses
j j
of locally compactAbelian groups (finite, discrete, discrete torsion, compact totally
disconnected, etc.) Since Gaussian distributions on a totally disconnected group are
degenerate, idempotent distributions play an important role on such groups. It turns
out that, in contrast to the classical situation, there are essential distinctions between
the cases when we deal with linear forms of two random variables, of three random
variables,orofn(cid:3)4randomvariables. Inthegroupsituationsomeneweffectsappear
whichdonotholdontherealline. Toshowthisconsiderthefollowingexample. Let
Z.5/bethegroupofresiduesmodulo5,and(cid:2) ,j D1;2;:::;n,n(cid:3)2,beindependent
j
random variables with values in Z.5/. If n D 2 and the linear forms L and L are
1 2
independent,thenallrandomvariables(cid:2) haveidempotentdistributions. IfnD3and
j
thelinearformsL andL areindependent,thenwecanonlyassertthatatleastone
1 2
oftherandomvariables(cid:2) hasanidempotentdistribution. Ontheotherhandforevery
j
n (cid:3) 4 there exist independent random variables (cid:2) , j D 1;2;:::;n, taking values
j
inZ.5/andautomorphisms˛ ,ˇ ofZ.5/suchthatthelinearformsL andL are
j j 1 2
independent,butall(cid:2) havenon-idempotentdistributions.
j
InSection13weconsidertwoindependentrandomvariables(cid:2) and(cid:2) . Weprove
1 2
thatifXisadiscretegroupandthelinearformsL andL areindependent,then(cid:2) and
1 2 1
(cid:2) haveidempotentdistributions. Wealsodescribecompacttotallydisconnectedgroups
2
forwhichthispropertyholdstrue. WeprovethattheSkitovich–Darmoistheoremfails
for compact connected groups. In Section 14 we study the case when the number n
ofrandomvariables(cid:2) ,j D1;2;:::;n,isgreaterthan2. Firstweassumethatthe(cid:2)
j j
takevaluesinafinitegroup. ThenwestudythecasewhenX iseitheracompacttotally
disconnectedgrouporadiscretetorsiongroup. Wedescribeinbothcasesthegroups
X forwhichindependenceofthelinearformsL andL impliesthateitherallrandom
1 2
variables(cid:2) haveidempotentdistributionsoratleasttwooftherandomvariables(cid:2) have
j j
idempotentdistributions,oratleastoneoftherandomvariables(cid:2) hasanidempotent
j
distribution. Furtherweconsideranarbitrarynumbern (cid:3) 4ofrandomvariables(cid:2) ,
j
andX isassumedtobeeitheradiscretetorsiongrouporacompactgroup. Wenotethat
our proof of the Skitovich–Darmois theorem for discreteAbelian groups in the case
Preface ix
when n D 2, in contrast to the proof of the Skitovich–Darmois theorem for discrete
Abeliangroupsinthecasewhenn>2,doesnotusetheUlmtheory.
In Section 15 we consider independent random variables (cid:2) and (cid:2) taking values
1 2
in an a-adic solenoid †a. We describe all possible distributions of random variables
(cid:2) assumingthatthelinearformsL andL areindependent. Theresultdependson
j 1 2
bothana-adicsolenoidandtopologicalautomorphisms˛ ;ˇ .
j j
In Chapter VI we study group analogues of the Heyde theorem. Let (cid:2) , j D
j
1;2;:::;n, n (cid:3) 2, be independent random variables taking values in a locally com-
pactAbeliangroupX. Let˛ , ˇ betopologicalautomorphismsofX satisfyingthe
j j
condition
.i/ ˇ ˛(cid:2)1˙ˇ ˛(cid:2)1 aretopologicalautomorphismsofX foralli ¤j:
i i j j
Let L D ˛ (cid:2) C(cid:4)(cid:4)(cid:4)C˛ (cid:2) and L D ˇ (cid:2) C(cid:4)(cid:4)(cid:4)Cˇ (cid:2) . Assume first that the
1 1 1 n n 2 1 1 n n
characteristic functions of the independent random variables (cid:2) do not vanish. In
j
Section 16 we prove that symmetry of the conditional distribution of the linear form
L givenL impliesthatall(cid:2) areGaussianifandonlyifX containsnoelementsof
2 1 j
order2. Thisresultcannotbeimproved. IfagroupX containselementsoforder2,
then the following natural problem arises: to describe all possible distributions of
independentrandomvariables(cid:2) takingvaluesinX andhavingthepropertythatthe
j
conditionaldistributionofthelinearformL givenL issymmetric. Weassumethat
2 1
onthegroupX thereexisttopologicalautomorphisms˛ ,ˇ satisfyingcondition(i).
j j
A simple example of such a group is the two-dimensional torus X D T2, and even
in this case the problem is very interesting. It turns out that the distributions which
are characterized by the symmetry of the conditional distribution of the linear form
L givenL areconvolutionsofGaussiandistributionsconcentratedonadenseone-
2 1
parametersubgroupofT2anddistributionssupportedinthesubgroupofT2generated
byelementsoforder2.
InSection17wedroptheassumptionthatthecharacteristicfunctionsofindepen-
dent random variables (cid:2) do not vanish. We first study the case when (cid:2) take values
j j
in a finite group and then (cid:2) take values in a discrete group X. In the case when X
j
isdiscreteandthenumberofrandomvariables(cid:2) is2,weproveinparticularthatthe
j
symmetry of the conditional distribution of the linear form L given L implies that
2 1
therandomvariables(cid:2) and(cid:2) haveidempotentdistributionsifandonlyifthegroup
1 2
X containsnoelementsoforder2.
In the appendix we study the Kac–Bernstein and Skitovich–Darmois functional
equationsonlocallycompactAbeliangroupsintheclassesofcontinuousandmeasur-
ablefunctions.
TheauthorwouldliketothanktheB.VerkinInstituteforLowTemperaturePhysics
andEngineeringoftheNationalAcademyofSciencesofUkraineforprovidingexcel-
lentfacilitiesforwritingthisbook.
March2008 G.M.Feldman