Exact Hausdorff measure on the boundary of a Galto

arXiv:0707.4358v1[math.PR]30Jul2007TheAnnalsofProbability2007,Vol.35,No.3,1007–1038DOI:10.1214/009117906000000629cInstituteofMathematicalStatistics,2007EXACTHAUSDORFFMEASUREONTHEBOUNDARYOFAGALTON–WATSONTREEByToshiroWatanabeUniversityofAizuAnecessaryandsufficientconditionforthealmostsureexistenceofanabsolutelycontinuous(withrespecttothebranchingmeasure)exactHausdorffmeasureontheboundaryofaGalton–Watsontreeisobtained.InthecasewheretheabsolutelycontinuousexactHaus-dorffmeasuredoesnotexistalmostsurely,acriterionwhichclassifiesgaugefunctionsφaccordingtowhetherφ-Hausdorffmeasureoftheboundaryminusacertainexceptionalsetiszeroorinfinityisgiven.Importantexamplesarediscussedinfouradditionaltheorems.Inparticular,Hawkes’sconjecturein1981issolved.Problemsofde-terminingtheexactlocaldimensionofthebranchingmeasureatatypicalpointoftheboundaryarealsosolved.1.Introduction.Aninterestinghistoryoftheclassicalproblemofdeter-miningtheHausdorffandpackingdimensionsandthentheexactHausdorffandpackingmeasuresoftheboundaryofasupercriticalGalton–Watsontreeisfoundinthepreviouspaper[46].Itwasinitiatedin1973bythethesisofHolmes[18],whosesupervisorandexaminerwereC.A.RogersandS.J.Taylor,respectively.Theauthor[46]completelysolvedtheproblemofdeter-miningtheexactpackingmeasureoftheboundaryofthetreebyfillingthecriticalgapintheproofofthetheoremofLiu[22],whichhadbeenpointedoutbyBerlinkovandMauldin[4].Berlinkov[3]independentlystudiedtheexactpackingmeasuresofhomogeneousrandomrecursivefractalsand,asacorollary,heobtainedananalogousresultunderacertainadditionalas-sumptiononthetree.However,itwasstatedwithoutpreciseproofandhecouldnotidentifytheexplicitvalueoftheexactpackingmeasureoftheboundary.UponanoutlineofHawkes[17],theauthor[46]definedaran-domsequence{Y(n)}forn≤0asY(−n):=μ(Bn),thatis,thebranchingReceivedApril2005;revisedDecember2005.AMS2000subjectclassifications.Primary60J80,28A78;secondary60G18,28A80.Keywordsandphrases.Galton–Watsontree,exactHausdorffmeasure,shiftself-similaradditiverandomsequence,boundary,branchingmeasure,dominatedvariation,b-decomposabledistribution.ThisisanelectronicreprintoftheoriginalarticlepublishedbytheInstituteofMathematicalStatisticsinTheAnnalsofProbability,2007,Vol.35,No.3,1007–1038.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.12T.WATANABEmeasureoftheballBnwithdiametere−nontheboundaryofthetreeanddiscoveredthatitisashiftself-similaradditiverandomsequenceonacer-tainextendedprobabilityspace.Itisakeyfactforsolvingthisoldproblem,whichenablesustousenewlimittheoremsforshiftself-similaradditiverandomsequencesdevelopedbytheauthor[44].Inthepresentpaper,weextensivelyemploylimittheoremsof“limsup”typeforthesequence{Y(n)}andfindanecessaryandsufficientconditionforthealmostsureexistenceofanabsolutelycontinuous(withrespecttothebranchingmeasure)exactHausdorffmeasureontheboundaryofaGalton–Watsontree.Itisrepre-sentedbythenondominatedvariationoftherighttailofthemartingalelimitofthebranchingprocess,equivalently,bythenondominatedvariationoftheintegratedfunctionoftherighttailoftheoffspringdistribution.SeeCorollary1.1.InthecasewhereanabsolutelycontinuousexactHausdorffmeasuredoesnotexist,wegiveacriterionwhichclassifiesgaugefunctionsφaccordingtowhetherφ-Hausdorffmeasureoftheboundaryminusacertainexceptionalsetis0or∞.SeeTheorem1.2.Theexplicitvalueofφ-HausdorffmeasureoftheboundaryisdeterminedforeachexampleinthreeadditionaltheoremsbyclosingtheseriousgapsintheproofsofLiu[21].SeeRemark1.4.Inparticular,Theorem1.3canbeappliedtoobtainupperandlowerboundsfortheexplicitvalueoftheexactHausdorffmeasureofahomoge-neousrandomrecursivefractalsuchasthelimitsetofMandelbrot’sfractalpercolationandthepathofaself-avoidingprocessontheSierpinskigasket.See[14]andtheexamplesofBerlinkov[3].Moreover,aconjectureofHawkes[17]in1981issolved.SeeTheorem1.6.Asisfoundintheconcludingre-marks,ourproblemofdeterminingtheexactHausdorffmeasureisnotyetcompletelysolved.However,itisrealizedthatthestudyoftheexceptionalsetΔdefinedby(1.7)belowwillleadtothecompletesolution.Inwhatfollows,denotebyRdthed-dimensionalEuclideanspaceandletR+=[0,∞).LetZ={0,±1,±2,...},Z+={0,1,2,...},N={1,2,3,...},anddenoteU=S∞n=0Zn+withZ0+=∅.Wedenotei∈Zn+by(ik)nk=1or(i1,i2,...,in).Fori∈Zn+⊂U,wedefine|i|=n.LetI=ZN+.Wedenotei∈Iby(ik)∞k=1anddefine,fori∈I,|i|=∞.Fori=(ik)nk=1andj=(jk)mk=1inU,wedefinei∗j∈Uasi∗j:=(i1,i2,...,in,j1,...,jm).Inparticular,wehave∅∗i=i∗∅=i.Wedefinei|n=(ik)nk=1fori∈U∪Iwithn∈Z+∪{∞}satisfyingn≤|i|.Weunderstandthati|0=∅.Wesaythati≤jinU∪Iif|i|=n≤|j|andj|n=i.Inthisorder,wedefinei∧j∈U∪Ifori,j∈U∪Iasi∧j:=max{k∈U∪I:k≤iandk≤j}.Wedefineametricd(i,j)fori,j∈Iasd(i,j):=e−|i∧j|.Then(I,d)isanultrametricspace.DenotebyB(I)theclassofallBorelsetsin(I,d).Fromnowon,let{Ni,i∈U}beZ+-valuedi.i.d.randomvariablesonaprobabilityspace(Ω,F,P).Inparticular,putN:=N∅.Weassume,toavoidthetrivialcases,thatthesupportofthedistributionofNisnotaone-pointset.Wedenotebyf(s):=P∞n=0pnsnEXACTHAUSDORFFMEASURE3theprobabilitygeneratingfunction(p.g.f.forshort)ofthedistributionofN,wherepn:=P(N=n)forn∈Z+.Letfn(s)bethenthiterationoff(s)withitself.Weassumethata:=E(N)1andE