Weak dependence for a class of local functionals of Markov chains on $\mathbb{Z}^d$

0.16 MB·English

by  Carlo Boldrighini

Weak dependence for a class of local functionals
of Markov chains on Zd
C. Boldrighini1, A.Marchesiello2, and C. Saffirio3
5
1
1 DipartimentodiMatematicaG.Castelnuovo,SapienzaUniversitàdiRoma,PiazzaleAldo
0 Moro5,00185Roma.
2 GNFM,IstitutoNazionalediAltaMatematica,PiazzaleAldoMoro5,00185Roma.
∗
l 2FacultyofNuclearSciencesandPhysicalEngineering,CzechTechnicalUniversityinPrague,
u DeˇcˇínBranch,Pohranicní1,40501Deˇcˇín†
J
3InstitutfürMathematikUniversitätZürich,Winterthurerstrasse190,CH-8057Zürich‡
0
3

]
h Dedicatedtothe90.thanniversaryofAcademicianYu. M.Berezansky
p
- Abstract
h
InmanymodelsofMathematicalPhysics,basedonthestudyofaMarkovchain
mat ehrt=yo{fh tth}et¥=s0toocnhaZsdti,coonpeecraatnorprroevsteribcytedpetrotuarbsautbivsepaacreguomfelonctsalafuconncttiroanctsioHnproenp--
M
[ dbowedwithasuitablenorm.Weshow,ontheexampleofamodelofrandomwalk
in random environment with mutual interaction, that the condition is enough to

3 proveaCentralLimitTheoremforsequences f(Skh ) ¥ , whereSisthetime
v { }k=0
shiftand f isstrictlylocalinspaceandbelongstoaclassoffunctionalsrelatedto
6 theHöldercontinuosfunctionsonthetorusT1. b
8
4
7
1 Introduction
0
.
1
Manyproblemsin Physicsand othersciences lead to considerMarkovchainson the
0
d-dimensionallattice Zd withlocalinteraction(see [15]). Thestatesof thechainare
5
1 randomfields h t = h t(x):x Zd ,t Z+ = 0,1,... , with h t(x) S, where S is
{ ∈ } ∈ { } ∈
: usuallyafiniteorcountableset.
v
i In many models, notably in the work of R.A. Minlos and collaborators(see, e.g.
X
[1, 3, 4, 6, 13, 14, 15] and referencestherein)one can prove,usually byperturbative
ar arguments,theexistenceofaninvariantmeasureP onthestatespaceW =SZd,andof
asubspaceoflocalfunctionsH L (W ,P ),invariantwithrespecttothestochastic
M 2
operatorT and such that for all F⊂ HM with zero average F P =0, we have, for
someconstantm¯ (0,1), ∈ h i
∈
(TF)(x ) m¯ F , x W . (1)
M
| | ≤ k k ∈
Here isasuitablenormonH ,whichisasaruleidentifiedwiththehelpofan
M M
k·k
expansioninanaturalbasis.
[email protected]
∗
†[email protected]
‡chiara.saffi[email protected]
1
Ifoneconsiderssumsoffunctionalsdependingonthespace-timefieldh = h ¥
{ t}t=0∈
W =SZd Z+, Z = 0,1,2,... , of the type (cid:229) T f(Sth ), where S is the time shift,
Sh = h ×¥ an+d f {isafunctio}nalwhichislocta=l0inspace,onecannotinbgeneralob-
{ t}t=1
tbainaCentralLimitTheorem(CLT)byrelyingonproperbtiessuchasstrongmixingand
thbelike[11],whichneedrequirementsthatmaynotapplyormaybedifficulttoprove
[7].Theaimofthepresentpaperistoestablishpropertieswhichholdintheframework
describedaboveandaresufficientfortheCLTtohold.
Themodelstowhichourdescriptionaboveappliesareofdifferentnature,andthe
space H is based on explicit constructions, so that it is convenientto work on the
M
exampleofaparticularmodel. Themodelthatwe considerhereisa randomwalkin
dynamicalrandomenvironmentwithmutualinteractionintroducedinthepapers[2,3]:
theMarkovchainh ,t Z ,describesthe“environmentfromthepointofviewofthe
t +
∈
randomwalk”,anobjectwhichplaysanimportantroleintheanalysisofrandomwalks
inrandomenvironment[12].
OurresultsareinspiredbyaclassicalresultontheCLTforfunctionalsofindepen-
dentvariablesbyIbragimovandLinnik[11](Th. 19.3.1).
Inthenextsectionwedescribethemodel,whichisaperturbationofanindependent
model,andpresentthemainfeatureswhicharerelevanttoouranalysis.In§3weprove
somepreliminaryresultsandinthefinalsection§4weproveourmainresults.
2 Description of the model
We consideraversionofthemodelstudiedin [2, 3], whichdescribesa discrete-time
random walk X Zd, d 1, t Z , evolving in mutual interaction with a random
t +
fieldx = x (x)∈: x Zd≥, with∈x (x) S= 1 . Thestate spaceisW =SZd, and
t t t
thespaceo{fthe"traje∈ctori}es"(or"histo∈ries")o{f±the}environmentxˆ = x : t Z is
t +
{ ∈ }
W =SZd Z+. Measurabilityisunderstoodwithrespecttothes -algebrageneratedby
×
thecylindersets.
b Thepair(X,x ),t Z is aconditionallyindependentMarkovchain[15], i.e., if
t t +
A W isameasurable∈set,wehave
⊂
P(X =x+u,x A X =x,x =x¯)
t+1 t+1 t t
∈ | (2)
=P(X =x+u X =x,x =x¯)P(x A X =x,x =x¯).
t+1 t t t+1 t t
| ∈ |
Ifxˆ W isfixed,thefirstfactorontherightof(2)definesthe"quenched"random
∈
walk,forwhichweassumethesimpleform
b
P(X =x+u X =x,x =x¯)=P(u)+e c(u)x¯(x), u Zd,x¯ W . (3)
t+1 t t 0
| ∈ ∈
Heree >0 isa smallparameter,P isa probabilitydistributiononZd andc is a real
0
function on Zd, such that P(u) e c(u) [0,1), u Zd. We also assume that P is
0 0
± ∈ ∈
evenandcoddinu,andthatbotharefiniterange. Byhomogeneityinspaceitisnot
restrictivetoassumeX =0.
0
FortherandomwalktransitionprobabilityP,withcharacteristicfunctionp˜ (l )=
0 0
l(cid:229) u=∈Z0d,Pa0n(du,)ieni(ol r,ud)ewrteoamsseuemtaettehcahtniitciaslnasosnu-mdepgtieonneriante[3,]i,.ew.,e|ap˜l0s(ol n)e|e=dt1haiftathnedFoonulyrieifr
coefficientsofthefunction 1 areabsolutelysummable.
p˜0(l )
2
The evolution of the environment is independent at each site, so that P(x
t+1
A X =x,x =x¯)isasumofproductsofthefactors ∈
t t
|
P(x (y)=s X =x,x =x¯)=(1 d )Q (x¯(y),s)+d Q (x¯(y),s) (4)
t+1 t t x,y 0 x,y 1
| −
where s= 1, Q ,Q are symmetric 2 2 matrices, Q has eigenvalues 1,m , m
0 1 0
(0,1),andQ± issuchthatQ Q =O(×e ). Inwords,ateachsitex Zd theevol|uti|o∈n
1 1 0
− ∈
isgivenbythetransitionmatrixQ ,exceptatthesitewheretherandomwalkislocated,
0
wherethetransitionmatrixisQ .
1
AnaturalprobabilitymeasureonthestatespaceW istheproductP =p Zd, with
0 0
p =(1/2,1/2).IfQ =Q (noreactionontheenvironment)P isinvariant.
0 0 1 0
The model just described was first considered in [3] both for the annealed and
quenched case. If there is no reaction on environment (i.e., Q =Q ) the CLT for
0 1
theannealedandquenchedasymptoticsoftherandomwalkwasobtainedinageneral
setting[8]. Anon-perturbativeresultwasobtainedin[9].
Thefieldh (x)=x (X +x),t Z isthe“environmentfromthepointofviewof
t t t +
the particle”. h : t Z isalso∈a Markovchainwithstate space W , andit canbe
t +
shown[6,9]th{atitis∈equiv}alenttothefullprocess(X,x ),i.e,forallT Z ,T 1,
t t +
giventhesequenceh ,...,h onecanreconstruct(X ,x ),...,(X ,x ),a∈lmost-su≥rely.
0 T 0 0 T T
ThestochasticoperatorT ontheHilbertspaceH =L (W ;P ),isdefinedas
2 0
(T f)(h¯)= f(h )h =h¯ , f H (5)
t+1 t
h | i ∈
where the average is w.r.t. the transition probability (3). By our assumptions T
preservesparityundhe·irtheexchangeh h .
→−
In H we introducea convenientbasis. As Q is symmetric, its eigenvectorsare
0
e =(1,1)ande =(1, 1)withcorrespondingeigenvalues1andm . Wedenotetheir
0 1
−
componentsase (s),sothate (s)=s,e (s)=1,s= 1,andset
j 1 0
±
F G (h¯)=(cid:213) e1(h¯(x))=(cid:213) h¯(x), G G, (6)
∈
x G x G
∈ ∈
whereGisthecollectionofthefinitesubsetsofZd,withF 0/ =1. F G :G G isadis-
creteorthonormalcompletebasisinH,andfor f H wewrite{f(h )=∈(cid:229) G }G fG F G .
∈ ∈
ForM>1thedensesubspaceH H isdefinedas
M
⊂
HM = f =(cid:229) fG F G : f M=(cid:229) fG M|G |<¥ . (7)
{ k k | | }
G G
HM equippedwiththenorm MisaBanachspace. As F G (h ) =1,wehave
k·k | |
f H f ¥ f M, f HM. (8)
k k ≤k k ≤k k ∈
MoreoverH isclosedundermultiplication.Infact,asitistosee,
M
F G F G =F G G , G G ′=G G ′ G ′ G ,
′ △ ′ △ \ ∪ \
sothatif f,g HM and f =(cid:229) G fG F G ,g=(cid:229) G gG F G ,wehave
∈
(cid:229) G G
fg M= fG gG M| △ ′| f M g M. (9)
k k | ′| ≤k k k k
GG
′
Inthepaper[3]ananalysisoftheexpressionofthematrixelementsofT andits
adjointT ,relyingontheirspectralpropertiesfore =0,leadstothefollowingresults.
∗
3
Theorem 2.1. If e and m are small enough, the space H is invariant under T,
M
andthereisaninvariant|pro|babilitymeasureP forthechain h whichisabsolutely
t
continuous with respect to P with uniformly bounded densi{ty v}(h ). Moreover H
0 M
canbedecomposedas
H =H(0)+H
M M M
where H(0) is the space ofthe constants, andon H the restriction of T actsas a
M c M
contraction:
T f M m¯ f M, fc HM, (10)
k k ≤ k k ∈
m¯ (0,1),m¯ = m +O(e ). Furthermoreif f = f +f,f H(0), f H ,then
∈ | | 0 0c∈ M ∈ M
f = f(h )dP (h )= f(h )v(bh )dP (h ). b c
0 0
Z Z
3 Preliminary estimates
WedenotebyPP theprobabilitymeasureonW = 1 Zd×Z+ generatedbytheinitial
{± }
distribution P , and by Mt1, 0 t t the s -algebra of subsets of W generated by
t0 ≤ 0 ≤ 1
{h t}tt1=t0. AsP isinvariant,PP isinvariantunbderthetimeshift.
b
Weconsiderfunctionals f whichdependonlyonthevaluesofthefieldattheorigin,
i.e., on the sequence of random variables h (0) ¥ . We set for brevity z =h (0)
{ t }t=0 t t
and z ={z t :t ∈Z+}∈W + ={±1}Z+. Mtt01, 0≤t0 <t1 will denotethe s -algebra
generatedbythevariables h (0) t1 ,whichisasubalgebraofMt1.
{ t }t=t0 t0
b
Byabuseofnotation, f(z )maydenoteafunctiononW oronW ,accordingtothe
+
circumstances, and similarly for the s -algebrasMt1, 0 t <t . We also write M
andMt forMtt andMtt,respbectively. t0 ≤b 0 1 t
Inwhatfollowsif f isafunctiononW weintroducethenotation f()M (h )=
0
G(f)(h ),h W . ThefollowinglemmaisasimpleconsequenceofThehore·m| 2.1i.
∈
b
Lemma3.1. Let f(z )beacylinderfunctiononW ,dependingonlyonthevariables
+
z ,...,z ,m 1. ThenG(f)(h ) H and
0 m 1 M
− ≥ ∈
b
G(f) C max fg (1+m )m, (11)
M≤ g 0,...,m 1 | | ∗
(cid:13) (cid:13) ∈{ − }
(cid:13) (cid:13)
wherem =M m¯(1+(cid:13)2m¯)a(cid:13)ndC>0isaconstant.
∗
Proof. As f deppendsonlyonz ,...,z itcanbewrittenintheform
0 m 1
−
f(z )= (cid:229) fgY g(z ) (12)
g 0,...,m 1
⊂{ − }
b b
wherethesumrunsoverthesubsetsof 0,...,m 1 ,andthefunctions
{ − }
Y g(z )=(cid:213) z t, g =0/, Y 0/(z )=1 (13)
t g 6
∈
b b
are called “Walsh functions”. The first assertion follows from the fact that for any
subsetg = t ,t ,...,t Z ,t <t <...<t ,wehave
0 1 k + 0 1 k
{ }⊂
Gg(h¯):= Y g|Mt0 ∈HM, h¯ ∈W . (14)
(cid:10) (cid:11)
4
Infact,ifrj=tk 1 j tk j, j=1,...,k,Gg canbewrittenas
− − − −
Gg(h¯)=F 0 (h¯) TrkF 0 ...Tr1F 0 (h¯), h¯ W , (15)
{ } { } { } ∈
i.e.,Gg isobtainedbysuccessive(cid:2)applicationsofT and(cid:3)ofthemultiplicationoperator
byF 0 . AsbothoperationsleaveHM invariant,Gg HM.
{ } ∈
MoreoverthefollowinginequalityisprovedintheAppendix
Gg M M|g|m¯[|2g|](1+2m¯)[|g|2−1] Cm |g|, (16)
k k ≤ ≤ ∗
where[]denotestheintegerpart,andC>0isaconstantwhichiseasilyworkedout.
·
Theproofofthelemmafollowsbyobservingthattheinequality(16)implies
khf(·)|M0ikM≤ C g∈{0m,..a.,xm−1}|fg |g 0,(cid:229)...,m 1 m ∗|g|. (17)
⊂{ − }
We denotebyˆ the probabilitymeasureinducedbyPP onW +. ˆ is stationary
withrespecttothetimeshiftonW : Sz = z ,z ,... .
+ 1 2
{ }
Thefollowingassertionisasimpleconsequenceofthepreviouslemma.
b
Lemma3.2. Undertheassumptionsofthepreviouslemma,ifm¯ issosmallthatm <1,
thentheprobabilitymeasureˆ onW iscontinuous. ∗
+
Proof. We need to prove that any point z (0) = z¯ ¥ W has zeroˆ -measure.
{ k}k=0 ∈ +
ConsiderthecylindersZ (z (0))= z =z¯ : j=0,1,...,n 1 ,whicharedecreasing
n j j
Z (z (0)) Z (z (0))andsuchtha{t Z (bz (0))= z (0) . T−he}probabilities
n+1 n n n
⊂ b ∩ { }
b b ˆ Z (z (0)) = 1 b n(cid:213)−1 1+bz¯ z (18)
n 2n j j
(cid:16) (cid:17) *j=0(cid:0) (cid:1)+ˆ
b
arecomputedbyexpandingtheinternalproductintermsofthefunctionsY g:
n 1
(cid:213) − 1+z¯jz j = (cid:229) Y g(z¯)Y g(z ), z¯ ={z¯j}nj=−01.
j=0 g 0,...,n 1
(cid:0) (cid:1) ⊂{ − }
b b b
Recallingthat Y g(z ) =1,wehave
| |
b
(cid:229) Y g(z¯)Y g(z ) (cid:229) Y g(z ) =
(cid:12)(cid:12)(cid:12)*g⊂{0,...,n−1} +ˆ (cid:12)(cid:12)(cid:12)≤g⊂{0,...,n−1}(cid:12)(cid:12)D Eˆ (cid:12)(cid:12)
(cid:12) b b (cid:12) (cid:12) b (cid:12)
(cid:12)(cid:12)= (cid:229) Y g M0 () (cid:12)(cid:12)P = (cid:229) (cid:12) Gg() P (cid:12).
| · ·
g 0,...,n 1 g 0,...,n 1
⊂{ − }(cid:12)(cid:10)(cid:10) (cid:11) (cid:11) (cid:12) ⊂{ − }(cid:12)(cid:10) (cid:11) (cid:12)
Thereforebytheinequality(cid:12)(16)therightside(cid:12)isboundedby(cid:12) (cid:12)
C (cid:229) m |g|=C 1+m ∗ n.
2ng 0,...,n 1 ∗ (cid:18) 2 (cid:19)
⊂{ − }
Henceifm <1,therightsidetendsto0asn ¥ ,whichprovesthelemma.
∗ →
5
Fromnowonweassumethatm <1.
∗
Wepasstoconsiderfunctionsforwhichtheexpansion(12)isinfinite,i.e.,g runs
over the collection g of the finite subsets Z+. The functions Y g :g g , are an
orthonormalbasisinL2(W +,ˆ 0), whereˆ 0=p Z+ istheprobab{ilityme∈asur}eonW +
correspondingtotherandomvariables z ¥ beingi.i.d. withdistributionp ( 1)=
{ k}k=0 ±
1. Thecorrespondingseriesiscalled“Fourier-Walshexpansion”[10].
2
A map F :W T1, where T1 =[0,1) mod1 is the one-dimensionaltorus, is
+
→
definedbyassociatingtoapointz W thebinaryexpansionx=0,a a ... [0,1],
+ 0 1
wtwiothbainta=ry1e−x2zpta,ntsi∈onZs,+b.utFitbiescnobomt∈eisnvinevretirbtilbelebeifcawueseexthcleuddeyathdeicsepqouinentsceosfsTu∈c1hhtahvaet
z = 1 for all t large enough. Such sequences are a countable set, which has zero
t
ˆ -m−easure,andalso,byLemma3.2,zeroˆ -measure.
0
UnderthemapF thebasisfunctionsY g gointothefunctions
y g(x)=(cid:213) f t(x), g g.
t g ∈
∈
wheref (x)istheimageofz ,t Z ,i.e.,
t t +
∈
1, 0 x< 1
f 0(x)= 1, 1≤ x<12
(cid:26) − 2 ≤
andfort 0,f (x)=f (2tx),where2txisunderstood mod1.
t 0
≥
If f L (W ,ˆ )then f˜(x)= f(F 1x) L2(T1,dx)andcanbeexpandedinthe
2 + 0 −
orthonor∈malbasis y g :g g ,withcoefficie∈nts
{ ∈ }
1
fg = f(z )dˆ 0(z )= f˜(x)y g(x)dx. (19)
ZW + Z0
AnaturalwayoforderingthebcollectiobngofthefinitesubsetsofZ ,whichplays
+
an importantrole in the theory, is obtainedby setting g =0/ and g = t ,t ,...,t ,
0 n 1 2 r
{ }
whererand0 t1<t2<...<trareuniquelydefinedbytherelationn=2t1+...+2tr.
≤
WecallWalshseriesboththeexpansion
¥
f(z )= (cid:229) fgY g(z )= (cid:229) fg Y g (z ), (20)
n n
g g n=0
∈
b b b
and the correspondingexpansionsof f˜(x). For the latter, an importantrole is played
byaparticularsetofpartialsums
2k 1
S 2k(f˜;x)= (cid:229) fgy g(x)= (cid:229) − fgny gn(x) (21)
g 0,1,...,k 1 n=0
⊂{ − }
forwhichitcanbeseen[10]that
b
S (f˜;x)=2k k f˜(y)dy, a =m2 k, b =(m+1)2 k (22)
2k k − k −
a
Z k
wheretheintegermissuchthata x<b .
k k
≤
Thefollowingresultisprovedin[10]. Werepeatithere,withashorterproofbased
onconditionalprobabilities.
6
Lemma 3.3. Let f˜(x)beaboundedfunction. ThenitsWalsh-Fouriercoefficients fg ,
givenby(19),satisfythefollowinginequality
w (f˜;2 n 1)
fg − − , n=max t:t g , (23)
≤ 2n+2 { ∈ }
(cid:12) (cid:12)
wherew (f;d )isthe(cid:12)mo(cid:12)dulusofcontinuityof f˜:
f(x) f(x)
w (f;d )= sup | − ′ |. (24)
d
x,x T1
|x−′x∈′|=d
Proof. Wehave
fg =*f(z )(cid:213)t∈g z t+ˆ 0 =*t∈(cid:213)g\{n}z t Df(z )z n|M0n−1E+ˆ 0.
b b
GoingbacktoT1,andsettingxn= a20 +...+an2−n1,aj= 1−2z j,wehave
f(z )z Mn =2n xn+2−n f˜(x)(1 2f (x))dx=
(cid:12)D n| 0E(cid:12) Zxn − n
(cid:12)(cid:12) b=2n xn+(cid:12)(cid:12)2−n−1 f˜(x) f˜(x+2−n−1) dx, (25)
Zxn −
(cid:2) (cid:3)
fromwhich,takingintoaccount(24),theinequality(23)followsimmediately.
TheresultsaboveallowustoprovetheanalogueofLemma3.1forfunctions f such
that f˜(x)= f(F 1x)isHöldercontinuous: f˜ Ca (T1),a (0,1). Inwhatfollows
−
ifg Ca (T1)wedenoteby g Ca thenorman∈dby g a the∈semi–norm
∈ k k k k
g(x) g(y)
kgka = sup | x −ya |.
x,y T1 | − |
∈
Lemma3.4. Let f beafunctiononW ,suchthat f˜ Ca (T1),a (0,1). Ifm¯ isso
+
smallthatk :=2 a (1+m )<1,thenG(f) H and∈thefollowing∈inequalityholds
− M
∗ ∈
G(f) M≤ 1Ca k kf˜kCa , (26)
(cid:13) (cid:13) −
whereCa >0isapositivecon(cid:13)(cid:13)stant.(cid:13)(cid:13)
Proof. If2k n<2k+1 theFouriercoefficientg intheWalshseries(20)issuchthat
n
max t gn ≤=k. Hence,asdw (f˜;d ) d a f˜ a ,theinequality(23)gives
{ ∈ } ≤ k k
fgn ≤ k21f˜+kaa 2−ka , 2k≤n<2k+1. (27)
(cid:12) (cid:12)
Thereforewehave (cid:12) (cid:12)
(cid:13)(cid:13)(cid:13)(cid:13)2kn+(cid:229)=12−k1fgn(cid:10)Y gn|M0(cid:11)(cid:13)(cid:13)(cid:13)(cid:13)M≤ |2|1f˜+||aa 2−ka 2kn+(cid:229)=12−k1(cid:13)(cid:13)(cid:10)Y gn|M0(cid:11)(cid:13)(cid:13)M.
(cid:13) (cid:13)
7
Observemoreoverthatthenumberofelementsofg isr = g =u 1whereu is
n n n n n
| | −
thenumberof"1"inthebinaryexpansionofn. Hence,bytheinequality(16)wefind
2kn+(cid:229)=12−k1 Y gn|M0 M ≤Cks(cid:229)=−01(cid:18)k−s 1(cid:19)m ∗s=C(1+m ∗)k−1,
(cid:13)(cid:10) (cid:11)(cid:13)
(cid:13) (cid:13)
which,as f0/ f ¥ ,togetherwith(27),implies(26).
| |≤k k
4 Weak dependence and the Central Limit Theorem
Inthepresentparagraphweproveourmainresultsforsumsofsequencesofthetype
f(Stz ),t=0,1,.... AsPP andthemeasureˆ inducedbyitonW +areinvariantunder
timeshift,thesequenceisstationaryindistribution.
Inbwhatfollows denotesanaveragewith respecttoˆ ,PP orP , accordingto
h·i
thecontext. Moreoverwedenotebyc,i=1,2,...,andsometimesbyconst,different
i
constantswhichdependontheparametersofthemodel.
Let f beaboundedmeasurablefunctiononW +with f ˆ =0,and
h i
n 1
S (z f)= (cid:229)− f(Stz ), n=1,2,.... (28)
n
|
t=0
b b
If f admitsaWalshexpansion(20)then(cid:229) g gfg Y g ˆ = f ˆ =0,sothat
∈ h i h i
f(z )= (cid:229) fgY g(z ), Y g(z )=Y g(z ) Y g() ˆ . (29)
− ·
g g,g=0/
∈ 6 (cid:10) (cid:11)
b b b b b b
Inwhatfollowswemakerepeateduse ofthefactthatif f isafunctiononW and
G(f):= f()M H ,then,byTheorem2.1, f(St+h )M =TtG(f) H .
0 M h M
h · | i∈ h · | i ∈
b
Theorem4.1. Let f beafunctiononW ,dependingonlyonz ,...,z ,m 1,and
+ 0 m 1
suchthat f ˆ =0. Thenthedispersionofnormalizedsums Sn(z |f) ten−ds,as≥n ¥ ,
h i √n →
toafinitenon-negativelimit b
¥
s 2= f2() +2(cid:229) f()f(St ) (30)
f · ˆ · · ˆ
t=1
(cid:10) (cid:11) (cid:10) (cid:11)
andtheseriesisabsolutelyconvergent.Moreover,ifs 2f >0,thesequence Sn√(zn|f) tends
weaklytothecenteredgaussiandistributionwithdispersions 2. b
f
Proof. Theproofofthetheoremisbasedontwobasicinequalities.
f(·)f(St·)|M0 M≤c1kfk2¥ m¯max{0,t−m+1}(1+m ∗)2m, (31)
(cid:13)(cid:10) (cid:11)(cid:13)
G(Sn) (cid:13) c2 f ¥ (1+m(cid:13))m, G(S2n) c3 f 2¥ m(1+m )m, (32)
M≤ k k ∗ M≤ k k ∗
(cid:13) (cid:13) (cid:13) b (cid:13)
where(cid:13)S2(z (cid:13)f)=S2(z f) S2( f) andc(cid:13),c ,c(cid:13)areconstantsindependentofm.
(cid:13) n |(cid:13) n | −h n ·| i (cid:13)1 2 (cid:13)3
b b b
8
Fortheproofof(31),observethatift m 1,then f(2)(z ):= f(z )f(St(z ))isa
t
cylinderfunctionM0t+m−1-measurablean≤dbou−ndedbykfk2¥ . Hence,byLemma3.1
and(19), ft(2) M const f 2¥ (1+m )m+t,and(31)holdsforbt m b1. b
Ift kmtakking≤theexpkectkationwit∗hrespecttoMm 1 wehave≤ −
≥ 0−
f()f(St )M = f(z )[Tt m+1G(f)](h )M .
0 − m 1 0
· · | − |
(cid:10) (cid:11) D E
AsG(f) H ,expanding f inWalshsebriesandusingProposition5.1intheAppendix
M
∈
andLemma3.1weseethatInequality(31)alsoholdsfort m 1.
≥ −
Thefirstcinequalityin(32)isasimpleconsequenceoftheinequality f(St )M =
0 M
TtG(f) M m¯t G(f) M,ofLemma3.1andoftheinequality fg kfh ¥ (s·ee| (19k)).
k k ≤ k k | |≤k k
Moreover,setting f2(z )= f2(z ) f2() and f(2)(z )= f(2)(z ) f(2)() ,wehave
t t t
−h · i −h · i
G(Sb2n) b n(cid:229)−1bT jG(f2) +2bn(cid:229)−2n−b(cid:229) j−1 T jGb(ft2) , (33)
M≤ M M
j=0 j=0 t=1
(cid:13) b (cid:13) (cid:13) b (cid:13) (cid:13) b (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
and the second inequality in (32) followsby observingthat f2 is a cylinder function
withzeroaverageand f2 ¥ f 2¥ ,andusingtheestimate(31)for G(ft2) M.
k k ≤k k k k
b
Passing to the assertions of the theorem, observefirst that, by the probperty(8) of
H ,theabsoluteconverbgenceoftheseriesin(30)followsfromInequality(31).
M
Assumingthats 2>0,fortheproofoftheCLTweadoptaBernsteinscheme. Let
f
p =[nb ],q =[nd ],k =[ n ],with0<d <b <1/4. Theintervalofintegers
n n n p(n)+q(n)
[0,n 1]isdividedintosubintervalsoflength p andq :
n n
−
I =[(ℓ 1)(p +q ), ℓp +(ℓ 1)q 1], J =[ℓp +(ℓ 1)q ,ℓ(p +q ) 1],
ℓ n n n n ℓ n n n n
− − − − −
ℓ=1,...k ,andtherestJ =[0,n 1] kn [I J ].
n ∗ − \∪j=1 j∪ j
Thesum(28)isthenwrittenasS (z f)=S(M)(z f)+S(R)(z f)where
n n n
| | |
S(M)(z f)= (cid:229)kn S (z f), bS(R)(z f)=b(cid:229)kn S (z f)b+S (z f) (34)
n | ℓ=1 Iℓ | n | ℓ=1 Jℓ | J∗ |
b b b b b
andS ,S ,S denotethesumsoverthecorrespondingsubinterval.
Iℓ Jℓ J
∗
WefirstprovethattheL -normofS(R)/√nvanishesasn ¥ ,i.e.,
2 n
→
2
(cid:229)kn S ( f) =k S2 ( f) +2 (cid:229) S ( f)S ( f) =O(n1 b +d ).
* ℓ=1 Jℓ ·| ! + n J1 ·| 1 s<t knh Js ·| Jt ·| i −
(cid:10) (cid:11) ≤ ≤ (35)
Fortheproof,observethatInequality(31)impliesthat S2 ( f) =O(q ),sothat
h J1 ·| i n
thefirsttermontherightof(35)isoftheorderk q n1 b +d .
n n −
∼
Forthesecondterm,observethat,bytranslationinvariance,recallingthatS (z f)
qn |
isMq0n+m−2-measurableandtakingthecorrespondingconditionalprobability,
b
hSJs(·|f)SJt(·|f)|M0i= Sqn(·|f) T(t−s)ℓn+pn−m+2G(Sqn) (h qn−m+2)|M0 , (36)
D h i E
9
whereℓ =p +q . Therefore,recallingtheinequalities(8),wegettheestimate
n n n
|hSJs(·|f)SJt(·|f)i|≤const(1+m ∗)mkfk2¥ qnm¯(t−s)ℓn+pn−m. (37)
Asknqnm¯pn constm¯ p2n,thedoublesumontherightof(35)isoftheorderO(m¯ p2n),
≤
sothat(35)isproved.
AsforS ,(31)implies S2 ( f) S2 ( f) =O(n 1+b ). Thisfact,together
with (35), pJr∗oves that (S(Rh)(Jz∗ ·f|))2i≤/nh=pnO+q(nn·|b +di), and,−as b >d , S(R) does not
n − n
h | i
contributetothelimitingdistribution.
Wenowshowthattherandbomvariables S kn arealmostindependentforlarge
{ Iℓ}ℓ=1
n,i.e.,forthecharacteristicfunctionsf (ℓ)(l z )=exp i l S (z f) wehave
n | { √n Iℓ | }
(cid:213)kn f (ℓ)(l z ) (cid:213)kn f (ℓ)(bl z ) 0, nb ¥ . (38)
n n
*ℓ=1 | +−ℓ=1 | → →
D E
b b
Weproceedbyiteration.Asafirststepweconsiderthedifference
(cid:213)kn f (ℓ)(l z ) k(cid:213)n−1f (ℓ)(l z ) f (kn)(l z ) =
n n n
*ℓ=1 | +−*ℓ=1 | + |
D E
b b b
= k(cid:213)n−1f (ℓ)(l z )f (kn)(l z ) , f (ℓ)(l z )=f (ℓ)(l z ) f (ℓ)(l z ) . (39)
n n n n n
*ℓ=1 | | + | | − |
D E
Weexpandf (knb)(lbz )inTbaylorseriebsatl =b0,wehave,bforsomel , bl l ,
n
| ∗ | ∗|≤| |
l l 2 l 3
f (kn)(l z )=bi S b(z f) S2 (z f) (S2 ( f) +i3 R (l ,z ),
n | √n Ikn | − 2n Ikn | − Ikn ·| n233! n ∗
(cid:16) D E(cid:17) (40)
b b b l b l b
Rn(l ∗,z )=SI3ℓ(z |f)exp{i√∗nSIℓ(z |f)}− SI3ℓ(z |f)exp{i√∗nSIℓ(z |f)} , (41)
(cid:28) (cid:29)
Clearly|Rnb(l ∗,z )|≤b2p3n|l |3kfk3¥ =bO(n3b ),sothbat,asb <1/4,webneedonlycon-
siderthefirsttwotermsoftheexpansion(40).
Theproductbofthefirstk 1factorsintheexpectationin(39)ismeasurablewith
n
respecttoMtn,wheret =(k− 1)p +(k 2)q +m 2. Takingthecorresponding
0 n n− n n− n −
conditionalexpectation,byInequality(32)wegetforthefirstordertermtheestimate
(cid:12)*kℓ(cid:213)n=−11f n(ℓ)(l |z ) Tqn−m+2G(Spn) (h tn)+(cid:12)≤c4m¯qn−m(1+m ∗)mkfk¥ . (42)
(cid:12) h i (cid:12)
(cid:12) b (cid:12)
(cid:12) (cid:12)
Fo(cid:12)rthesecondorderterm,proceedinginthe(cid:12)sameway,andtakingintoaccountthe
secondinequality(33)wecometotheestimate
(cid:12)*kℓ(cid:213)n=−11f n(ℓ)(l |z ) Tqn−m+2G(S2pn) (h tn)+(cid:12)≤c5m¯qn−mm(1+m ∗)mkfk2¥ . (43)
(cid:12) h b i (cid:12)
(cid:12) b (cid:12)
(cid:12) (cid:12)
I(cid:12)terating the procedure for the remaining p(cid:12)roducth(cid:213) ℓk=n−11f n(ℓ)(l |z )i, we see that
thequantityontheleftof(38)isoftheorderO(knn−3(21−b ))=O(n−21+2b ), so that,
asb <1/4,itvanishesasn ¥ . b
→
10