Functional central limit theorems for

Charles Suquet

Description

Functional central limit theorems for self-normalized partial sums of linear processes Alfredas Ra£kauskas and Charles Suquet July 10, 2010 Abstract We prove the invariance principle under self-normalization by blocks for linear processes with summable filters and i.i.d. innovations in the domain of attraction of the normal distribution. Keywords: Domain of attraction, invariance principle, linear processes, self-normalization, weak convergence. 1 Introduction and results We consider linear processes Xk = ∑ i?Z aik?i, k ? N, (1) where (ai, i ? N) is a square summable ( ∑ i?Z a 2 i <∞) sequence of real numbers and (i, i ? Z) are i.i.d. centered random variables in the domain of attraction of the normal law (writen 1 ? DAN). This implies in particular that E |i|p <∞ for each 0 < p < 2 and, consequently Xk is well defined (see, e.g., Brockwell and Davis [4]). Central limit theorem for partial sums Sn = X1 + · · · + Xn, n ? N, and functional limit theorems for processes build from partial sums (Sk, k ? N) has been extensively studied in the literature. We refer to the survey paper by Merlevède, Peligrad and Utev [15] for recent results on the central limit theorem and its weak invariance principle for stationary sequences under finite second moment assumption.

• dependent variable


• self-normalized partial


• convergence


• functional central


• n?∞


• banach space


• u?1n ?n


• central limit


• centered random

Sujets

• Suquet
• Tamboril
• Dependent and independent variables
• Convergence
• Banach space

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 10
• Langue: English

Extrait

X
X = a ; k2N;k i k i
i2Z
P 2(a ;i2N) a <1i i2Z i
( ;i2Z)i
p 2 DAN) Ejj <11 i
0<p< 2 Xk
S =X + +X ;n2N;n 1 n
(S ;k2N)k
i
Vn
2 2 2V =X + +X :n 1 n
1V Snn
X ;X ;:::1 2
cesses,oalsodistributi[5].normalariance.thecaseofHahnattractionassumption.ftheotoisauskaehasquarewsforumnotmableis(attraction,domain(2)hesequencetInin,ationsortanvrinnoori.i.d.gandin)nitesequenceerofords:realhnhnique.umDomainbprinciple,ersproandliteratureltersehasummableductionwithtcessesedouerwiczpelinearofareizedi.i.d.LogancenandteredinrandomKl?ppvhariablessurvinariancethesequencesdmomenothismaineofthatattractionniteofsthetextnormalappropriatelathiswb(writen,forvksinearcwbloconsideryric.devThilimitsofimpliesresultsininparticularindepthatvb1self-normalizationRa?underandprinciple[18,arianceSzyszkvWinortheimforaspeaimitcofhweferval.proZhangetherein.Wtecand,seriesconsequenrefertlyeAbstractMik2010Wistowpapellvdenedprinciple(see,stationarye.underg.,secondBrotcInkwpapellwandshallDaassumevisKeyw[4]).hasCenvtralInlimituctheoremconforself-normalizationpartialansumstec10,UsuallyJulymeansSuquetnormalizationCharlesyandofaswhereauskinkarianceRa?lAlfredasprocessesself-normalization,processeslinearlinearofesumsApartialhself-normalizedis.otedwherethearticularlybtheviortralthetheoremWcompletelyandedtroytheG?tzofandenden[8]randomiariablesvn(1)ergence.vprinciplesconproandPfunctional,limitcentheoremslimitforisprosolvcessesbbuildGin?,frome,partialMasonsumsandforntheoremsarianceeakwerelimitvtralincenkunctionalasFSuqbtde19]PCs?rg?,KlassoLaiand1anginFtheothliterature.rWpetreferectstolthebsuviourrvself-normaleysumspapeerebtoyetMerle[14],vand?de,[9]PreferenceseligradFandself-normalizationUtevhniques[15]timeforanalysis,recenettoresultselbonrtheandcenosctral[12].limitetheoremreferandaitseywereakyhaslabe?a,eenandextensiv[6]elystudiedB Xm;j k
N = [n=m] m =m(n)
jmNX X
2 2U = B ; B = X :m;j in m;j
j=1 i=(j 1)m+1
2 DAN1
b "1n
nX
1b ! N(0; 1); :kn n!1
k=1
P P
jaj<1 a = 0 2 DANi i 1i2Z i2Z
1U S! N(0; 1);nn
n!1
m!1 m=n! 0 n!1
2 DAN1
a = 1 a = 0 i = 0 X =0 i k k
1
P
jjaj <1:jj
[nt]X
(t) = X + (nt [nt])X ; t2 [0; 1]n i [nt]+1
i=1
C[0; 1]
[0; 1]
jjxjj = sup jx(t)j:
t2[0;1]
f (t) :t2 [0; 1]gn
[nt]X
(t) = Xn i
i=1
W =fW (t) :t2 [0; 1]g [0; 1]
D
!
onloofokingwingatethe(see,sptheecialraccaseawheretivthetfunctionsendenwithdepalsoand6inincessesorohoproleself-normalizededforcesses.ofwnian6ofapplications(5)andin,rew(7)elhanormalvresequencetheoryfollothetof1]resultshtimitsandthethenorohothe14].memtheoremsbdenoteershipisofthatrecentinforovideinedindicatednormOur,reLetulthatsthentheewingstepwprotheorems.ofMasondistribution)[8].tEarlierbthe,conhv(4)ergence(8)(5)elewtheasspaceobtainedallb1]yvJuot-handdisareanduousRa?t,ktheausktopas[2,[10]1.underlinearthetralstrongertributionconditionstandardIndeedon1..Theorem,indconsideredconltersuousofonclassprwholeequippthewithTheoremuniform1distribution,iswhereactuallydenea(3)corollaryusofmindfunctional(domainlimitattt,heWoremsconsiderprothevpartiaedsumsincessthistionpaptheonandwmeansdenotethatconheergencedeneddistributionyolygonalexistslineaprosuccessthatholddistributiontoThe(5)IfergenceasvrandomconmentheofforSknecessarydisD[0,ofariables.functionsW[0,ewhicconsiderhaself-normalizationefconditionltheandClearlycon.nusingfromblorighcequippaswithk-sumsSkofd's.ologyande.g.ChoSection(6)LetandTheoremvieprowforitlimitascenatorandomconelementhetBroinmotiontheourBanactheoremhByspaceosinger.eWtheevdeneintheptheinspace.DmainANsistnecessaryarebfolloytGin?,oG?tze,2andP P
jaj<1 a = 0 2 DANi i 1i2Z i2Z
D1U ! W C[0; 1];nn
n!1 m!1 m=n! 0 n!1
P P
jaj<1 a = 0 2 DANi i 1i2Z i2Z
D1U ! W D[0; 1];nn
n!1 m!1 m=n! 0 n!1
ai P P1 1 2 1=2 V =j aj ( a )n in i i i
Un
V n
m =m(n)
() ()
V () U () n n n n
X
2 DAN (b )1 n
> 0
nP (jj>b! ) 0;1 n
n!1
n 2E 1 ! 1;fjj b g1 1 n2 n!1bn
nX
Pr2 2b ! 1:n k n!1
k=1
2 DAN (b )1 n n1
n
E jj1 ! 0:1 fjj>b g1 n n!1bn
Z 1n n
E jj1 =nP (jj>b ) + P (jj>x) dx:1 fjj>b g 1 n 11 nb bn n bn
P (jj>x)1
2L(x) :=E 1 2 DANfjj xg 11 1
(11).acutey(10)casdothe2obanjetermctsedenedinsteadsubstitutinginformationcessbbassumptionsyIfproinint(2),same(t3truncated)linear,D(8),e(6)orespeectivvelyce.ortanIf,theinforthisresultoximationtoapprov,forthencowithknotheinnormalizingissequencethestrongselfnormalizationapropobtainedout[13]hereasfactsin(14)(4),InoneparthasreacforseemseacabhsoKulikc's,problem.,an,6profthesituations,outOfabecienassumptionrssamethetheeUsingside.bovideodththatvofsecondterestitasvandexpress(11)needinw,ofandmomen6sptheeations,withapplicandstatistical,ofertviewANInthe.ab(9)someasgather,WprUsefulovideh.dPr,of.,tegratingthatyandw(12)obtainIfofTheoremo3eandoGin?as[1],generalChap.under2,hoiCor.optimal4.8(a),ObtainingTh.t6.17imp(i)isand,Cor.and6.18then(b).hoiceMoreothevpracticalercourse3..Theoremt.coasTheytbindenoteabevwrighconfusion,handtationaltendsno-zerooidyvTaprooe(13)eLemmacon4.ergenceIftheTttheorems.rm,limitisournofenienwithtonormalizingwsetoquencnoteeofsthatprotermsthetheinsecondusedtcesses,theproofasac2.yinbsp,,thenSincein.(4)self-normalization,thethensee,e.g.Araujo2L x P (jj >1
x) =o(L(x)) x!1
L(x)g(x)
P (jj>x) = ; x> 0;1 2x
g g(x) 0
g [c;1)
> 0 x L(x)
2 (0; 1) n0
nn b c0 n
L(x) 2L(b )n
; xb :n x bn
Z Z Z1 1 1n 2nL(b ) g(x) 2nL(b ) g(b t)n n n
P (jj>x) dx dx = dt:1 1+ 2 2 2 b x b tbn b n b 1n n n
0
g [c;1)
2 DAN1
2 V () [nt]