A Bernstein Inequality For Exponentially Growing Graphs

0.14 MB·

by  Johannes T. N. Krebs

A Bernstein Inequality For Exponentially Growing Graphs
∗
JohannesT.N. Krebs†
January17, 2017
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J Abstract

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1 In this article we present a Bernstein inequality for sums of random variables which are defined on a graphical

] networkwhosenodesgrowatanexponential rate. Theinequalitycanbeusedtoderiveconcentrationinequalitiesin
T
highly-connectednetworks.Itcanbeusefultoobtainconsistencypropertiesfornonparametricestimatorsofconditional
S
. expectationfunctionswhicharederivedfromsuchnetworks.
h
t
a Keywords: Asymptotic inference; asymptotic inequalities; Bernstein inequality; Concentration inequality; Graphs;
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Highly-connectedgraphicalnetworks;Mixing;Nonparametricstatistics;Randomfields;Stochasticprocesses
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MSC2010: Primary:62G20,62M40,90B15;Secondary:62G07,62G08,91D30
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v
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1 InequalitiesoftheBernsteintypeareanimportanttoolfortheasymptoticanalysisinprobabilitytheoryandstatistics.
4 TheoriginalinequalityderivedbyBernstein(1927)givesboundson (S > ε),whereS = n Z forbounded
0 P | n| n k=1 n
random variables Z ,...,Z which are i.i.d. and have expectation zero. There are various versions of Bernstein’s
. 1 n P
1
inequality,e.g., Hoeffding(1963). Inparticular, generalizationsto differentkindsof stochastic processeshave gained
0
importance: Carbon(1983), Collomb(1984), BrycandDembo(1996) andMerleve`deetal. (2009) provideextensions
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1 to times series Z : t which are weakly dependent. Valenzuela-Dom´ınguezandFranke (2005) give a further
t
v: generalization t{o strong∈miZxi}ng random fields Zs : s N which are defined on the regular lattice N for some
{ ∈ Z } Z
i lattice dimensionN . Thecorrespondingdefinitionsofdependenceandtheirinteractionpropertiesaregivenin
X ∈ N+
Doukhan(1994)andinBradley(2005).
r
a Bernsteininequalitiesinparticularfindtheirapplicationswhenderivinglargedeviationresultsor(uniform)consistency
statementsinnonparametricregressionanddensityestimation,compareGyo¨rfietal.(1989)andGyo¨rfietal.(2002).
InthisarticlewederiveaBernsteininequalitywhichadaptstohighly-connectednetworkswherethenumberofnodes
growsatan exponentialrate. A well-knownexampleforsucha graphis theinternetmapwhichtries to representthe
internetwithvisualgraphics. Anotherapplicationmaybenestedsimulationswhichareusedininsurancemathematics
to simulate the outcomeof an insurancecontract. We deriveanotherconcentrationinequalitybased on thisBernstein
inequality.Itturnsoutthatweneedasomewhatstricterdecayintheα-mixingcoefficientsthanitisusuallyassumedin
thecasefortimeseries.
Thispaperisorganizedasfollows: wegivethemotivationandthedefinitionsinSection1. Section2containsthenew
Bernsteininequalityandconcentrationinequalitiesforexponentiallygrowinggraphs,itisthemainpartofthisarticle.
TheAppendixAcontainsausefulpropositionofDavydov(1973).
∗ThisresearchwassupportedbytheFraunhoferITWM,67663Kaiserslautern,GermanywhichispartoftheFraunhoferGesellschaftzurFo¨rderung
derangewandtenForschunge.V.TheauthorthanksHannesChristiansenforproofreadingpartsofthearticle.
†DepartmentofMathematics,UniversityofKaiserslautern,67653Kaiserslautern,Germany,email:[email protected]
1
1 Introduction
InthissectionweconsiderageneralgraphG=(V,E)withacountablesetofnodesV andasetofedgesE. Wedefine
thenaturalmetriconGastheminimalnumberofedgesbetweentwonodes
d :V V ,
G
× →N
(v,w) inf l suchthatthereare(v ,v ),...,(v ,v ) E with v =v,v =w .
0 1 l−1 l 0 l
7→ ∈N ∈
(cid:8) (cid:9)
Themetricd isextendedtosetsI,J V intheusualway: d (I,J) = inf d (v,w) : v I,w J . Wedenote
G G G
⊆ { ∈ ∈ }
byN(v) the setofneighborsofv w.r.t.Gfora nodev ofa graphG = (V,E). Furthermore,we assumethatthereis
aprobabilityspace(Ω,A, )whichisendowedwithareal-valuedrandomfieldZ. Thelatterisindexedbythesetof
P
nodesV, i.e., Z isafamilyofrandomvariables Z : v V suchthatZ : Ω S ismeasurableforeachv V.
v v
{ ∈ } → ∈
Wedenotetheindicatorfunctionby andwedefinetheα-mixingcoefficientoftherandomfield Z : v V onthe
v
1 { ∈ }
graphG=(V,E)by
α (n):= sup sup (A B) (A) (B), n .
G
I,J⊆V, A∈F(I),|P ∩ −P P | ∈N
dG(I,J)≥nB∈F(J)
Therandomfieldisstrongmixingw.r.t.Gifandonlyifα (n) 0forn . Inthesequel,weinvestigaterandom
G
→ →∞
fieldswhicharedefinedonthefollowingclassofgraphs:
Definition1.1(TreesgrowingatanexponentialrateA). LetA . AtreeT =(V,E)isgrowingatanexponential
+
∈N
rate Aif T is a rootedtreeandeach nodev V hasexactlyA children. Thenodesin the tree arelabeledaccording
∈
tothefollowingscheme: thedistinguishedroot(whichhasnoparent)islabeledby(0,0)andthechildrenofthenode
(j,k)arelabeledby(j+1,A(k 1)+1),...,(j+1,Ak). Hence,thesetofnodesandthesetofedgesaregivenby
−
V = (j,k):j , 1 k Aj andE = ((j,k),(j+1,k′)): (j,k) V andA(k 1)+1 k′ Ak .
∈N ≤ ≤ { ∈ − ≤ ≤ }
(cid:8) (cid:9)
ArootedgraphG=(V,E)isgrowingatanexponentialrateAiftheedgesEcanbedecomposedintotwodisjointsets
asE = E′ E˜ suchthat(V,E′)is atree growingatan exponentialrateAandthesetE˜ of additionaledgeshasthe
∪
propertythatitdoesnotconnectnodesofarbitrarylengthinT,i.e.,
sup d (v,w) (v,w) E˜ < .
T
{ | ∈ } ∞
We come to the definition of an embeddable graph. Here it is worthy to mention that especially in the context of
−
graphtheory therearedifferentdefinitionsofgraphembeddings:thecommondefinitionofanembeddingofagraph
−
G requires, loosely speaking, that the edges of the embedded graph may only intersect at their endpoints, i.e., at the
nodes. Itiswellknownthatanygraphwithcountablymanynodescanbeembeddedinto 3 viaplacingthei-thnode
Z
atthepoint(i,i2,i3) 3,compareCohenetal.(1994). Furthermore,onecancharacterizethefinitegraphswhichare
∈Z
embeddableintotheplane(theplanargraphs)withthehelpofthetheoremsofKuratowski(1930)andofWagner(1937).
Here,weslightlychangethisgraphtheoreticdefinitionsuchthatitistailoredtoourneeds: wecanomittherestriction
thatedgesmaynotintersectataninteriorpoint.However,sinceweshallusuallybedealingwithinfinitegraphs,wehave
toaddarequirementthatisessentialwheniscomestomixingrandomfieldswhicharedefinedonthegraphwhichisto
beembedded.Weneedthisdefinitiontoshowwhatisintuitivelyclear:theBernsteininequalitiesforregularlatticesare
notapplicableinthecontextofgraphswhichgrowatanexponentialrate. Wegivethedefinition
Definition1.2(EmbeddableGraphs). LetG=(V,E)beagraphwithcountablymanynodesV anddenotebyd the
p, N
Euclideanp-normontheN-dimensionallattice N,forp 1.Gisembeddablein N ifthereisadimensionNZ
+
Z ≥ Z ∈N
suchthatGisisomorphictoagraphG′ =(V′,E′)withV′ N andforeachsequence((v ,w ) : i ) V V
i i
⊆Z ∈N ⊆ ×
withimage((v′,w′):i ) V′ V′itistruethat
i i ∈N ⊆ ×
sup d (v ,w ):i < = sup d (v′,w′):i < .
{ G i i ∈N} ∞ ⇒ { ∞,ZN i i ∈N} ∞
2
Inthefollowing,whenspeakingofthelattice N asa graph,we shallalwaysunderstandthe graphG = (V,E)with
Z
nodesV = N andedgesE = (v,v+b ):v V, i=1,...,N whereb isthei-thstandardbasisvectorwhichisone
i i
Z { ∈ }
inthei-thcoordinateandzerootherwise.Notethatinthiscase,wehaved d andd d Nd .
G ≡ 1,ZN ∞,ZN ≤ 1,ZN ≤ ∞,ZN
Wehaveapracticallemmawhichgivesequivalentformulationsofthisdefinition
Lemma1.3. LetGbeagraph.Thenthefollowingareequivalent
1. Gisembeddablein N
Z
2. GisisomorphictoagraphG′ =(V′,E′)withnodesV′ N andthereisaconstant0<C < suchthatfor
⊆Z ∞
any(v,w) V V withimage(v′,w′) V′ V′itistruethatd (v′,w′) Cd (v,w).
∈ × ∈ × ∞,ZN ≤ G
3. GisisomorphictoagraphG′ =(V′,E′)withnodesV′ N and
⊆Z
sup d (v′,w′): v V,w N(v),v(resp. w)isisomorphictov′ (resp.w′) < .
{ ∞,ZN ∈ ∈ } ∞
Inparticular,let Z :v V bearandomfieldonG,denoteby Z :s V′ thesamerandomfieldunderthegraph
v s
{ ∈ } { ∈ }
isomorphism. Thenthemixingcoefficientssatisfyasymptoticallyα ( C ) α whichmeansthatstrongmixing
mixingisinheritedwhenswitchingbetweenGandG′. ∞,ZN ⌈ ·⌉ ≤ G
Proof. (1) (2)and(3):letGbeembeddablein N,thenobviouslyV iscountable,thus,thenumber
⇒ Z
C :=sup d (v,w):d (v,w)=1,v,w V
{ ∞,ZN G ∈ }
ismeaningfulandfinitebyassumption.Consequently,wehavefortwoconnectednodesvandwinV thatd (v,w)
∞,ZN ≤
Cd (v,w). Ifvandwarenotconnectedthend (v,w)= . Hence,Cistheproperconstant.Theconverseinclusions
G G
∞
(2)(resp.(3)) (1)areimmediate.
⇒
We come to the amendment of the lemma. Let n be given and consider a random field on G and its graph-
∈ N
isomorphiccounterpartonG′. WeinferfortwosetsI′,J′ V′ withpreimageI,J V andd (I′,J′) nthat
Cd (I,J) d (I′,J′) n,i.e.,wehaveusingthegr⊆aphisomorphismforn ⊆C ∞,ZN ≥
G ≥ ∞,ZN ≥ ≥
(I′,J′): I′,J′ V′andd (I′,J′) n (I,J): I,J V andd (I,J) C−1n .
{ ⊆ ∞,ZN ≥ }⊆{ ⊆ G ≥ }
Thus, α (n) α C−1n for n C. This means that asymptotically α α C−1 or rather
∞,ZN ≤ G ≥ ∞,ZN ≤ G ·
α ( C ) α .
∞,ZN ⌈ ·⌉ ≤ G (cid:0)(cid:4) (cid:5)(cid:1) (cid:0)(cid:4) (cid:5)(cid:1)
Thefollowingclassofgraphscannotbeembeddedin N
Z
Proposition1.4(Exponentiallygrowinggraphsarenotembeddable). LetG=(V,E)beagraphwithrootv V. Put
0
∈
L := v andrecursively
0 0
{ }
L := v V k−1L w L withw N(v)
k ∈ \∪i=0 i |∃ ∈ k−1 ∈
(cid:8) (cid:9)
fork . Ifthemap k L growsfasterthananypolynomialfunctionofdegreeN definedon ,Gisnot
+ k
∈ N N ∋ 7→ | | N
embeddablein N.
Z
Proof. Letthemap k L growfasterthananypolynomialofdegreeN andassumethatGcanbeembedded
k
N ∋ 7→ | |
into N with0 <C < whichsatisfiesd (v,w) Cd (v,w)asstatedinLemma1.3. First,observethatforv
Z ∞ ∞,ZN ≤ G
andwbothinL thedistanceinthegraphisatmostd (v,w) 2k. Byassumptionthereisak suchthatforall
k G 1
≤ ∈N
k k wehave L > (2k+1)N. Thus,fork k therearev,w L withthepropertythatd (v,w) > 2k
≥ 1 | k| ≥ 1 ∈ k ∞,ZN
whichimpliesforthesetwonodesthat
2k<d (v,w) Cd (v,w) 2Ck.
∞,ZN ≤ G ≤
3
Hence,C >1. Inthesameway,thereisak suchthatforallk k ,wehave L >(2C2k+1)N. Inparticular,
2 2 k
∈N ≥ | |
thereare(v,w) L ,k k withthepropertythatd (v,w)>2C2kwhichimpliesfork max(k ,k )
∈ k ≥ 2 ∞,ZN ≥ 1 2
2C2k<d (v,w) Cd (v,w) 2Ck;
∞,ZN ≤ G ≤
whichinturnimpliesC <1. ThiscontradictstheassumptionthatGisembeddablein N.
Z
ThisimpliesthatwecannotusetheabovementionedBernsteininequalitiesfordatawhichisdefinedonalatticetoderive
concentrationinequalitiesforrandomfieldsthataredefinedongraphswhichgrowatanexponentialrateA. Insteadwe
giveanewBernsteininequalitywhichcandealwiththisclassofrandomfieldsinthenextsection.
2 A Bernstein inequality for exponentially growing graphs
In this section we derive inequalities of the Bernstein type for random fields which are highly-connectedand whose
indexsetgrowsatanexponentialrate. Weneedthefollowingimportantlemma:
Lemma2.1. LetT =(V,E)beatreegrowingatanexponentialrateA. Denoteby
V(j,k,P):= (j′,k′) V j j′ j+P 1, Aj′−j(k 1)+1 k′ Aj′−jk (2.1)
∈ | ≤ ≤ − − ≤ ≤
n o
thesetofnodesofthesubtreeofT whichhasitsrootatthenode(j,k)andconsistsofP generations. Consider
+
∈ N
the graph which is induced by the set of nodes V(j,k,P). Then the number of pairs (v,w) in this graph which are
separatedbyexactlyLedgesfor1 L 2(P 1)isgivenby
≤ ≤ −
⌊P−1−L/2⌋ L∧(P−1−h)
N(P,L):= Ah Ai 2 1 +(A 1)AL−i−1 L>i (2.2)
{L=i}
· − 1{ }
h=0 i=1∨(L−(P−1−h))
X X (cid:0) (cid:1)
AP AL
=2 − 1 +(L 1) AP AL−1 L P
A 1 {L≤P−1} − − 1{ ≤ }
−
+ 2(P 1) L+1 AP−1+⌊L(cid:0)/2⌋ AL−1∨(cid:1)P L 4
− − − · 1{ ≥ }
(cid:0) (A 1)(P 1 (cid:1)(cid:16)L/2 ) 1 AP−1+⌊L/2(cid:17)⌋+AL
2 − − −⌈ ⌉ − L 4
− A 1 · 1{ ≥ }
(cid:8) − (cid:9)
(A 1)(P L) 1 AP +AL
+2{ − − − } 4 L<P
A 1 1{ ≤ }
−
CP AP+L/2,
≤
forasuitableconstant0<C < whichdoesnotdependonP,LandA.
∞
ProofofLemma2.1. Theminimaldistanceinthissubtreeclearlyis1,whereasthemaximaldistanceis2(P 1). Let
−
nowalengthLbefixed,1 L 2(P 1). Wedistinguishtwocasesforapair(v,w)whichisseparatedbyLedges:
≤ ≤ −
inthefirstcasew(resp. v)isadescendantofv(resp. w). Inthesecondcasevandwhaveacommonparentwhichwe
callrand,plainly,v =r =w.
6 6
The first case is onlypossible for 1 L P 1, for such an L thereare exactly 2(AP AL)/(A 1)such pairs
≤ ≤ − − −
(v,w)inthissubtree.Thesecondcaseispossiblefor2 L 2(P 1).DependingonLtheparentislocatedbetween
≤ ≤ −
generationzeroandgeneration P 1 L/2 ,denoteitsgenerationbyh. Havingfixedaparentringenerationhthe
⌈ − − ⌉
distancefromrtothefirstnodevisatleast1 (l (P 1 h))andatmostL (P 1 h),denotethisdistanceby
∨ − − − ∧ − −
i. Hence,thereareexactlyAi nodesinquestionforv. Inthiscasethati<LthenodewisseparatedL igenerations
−
fromr. Sincev =wandtheirgraphdistanceisL,thisyields(A 1)AL−i−1possibilitiesforw. Allinall,wegivethe
6 −
numberofpairswiththeformulafromequation(2.2).
It followsthe Bernstein inequality. Here we do not considerthe full set of nodesV instead we focuson a strip of V
whichisdefinedwiththehelpoftheV(j,k,P)fromthepreviousLemma2.1.
4
Theorem2.2(Bernsteininequality). LetT =(V,E)beatreegrowingatanexponentialrateA.LetZ beareal-valued
v
randomvariable for eachv V with [Z ] = 0, Z C andVar(Z ) σ2, for some 0 < σ,C < . Let
∈ E v k vk∞ ≤ v ≤ ∞
L ,P andconsiderthesubtreeinducedbythesetofnodes
+
∈N ∈N
V′ :=V(L,1,P) ... V(L,AL,P) (2.3)
∪ ∪
withV(L,i,P)asinthedefinitiongivenin(2.1). Then
Z >ε 2e−βεexp 10√eα (f)(P2+Q2)/(2P2+2Q2+AL) AL
v T
P (cid:12)(cid:12)(cid:12)vX∈V′ (cid:12)(cid:12)(cid:12) !≤ (cid:26) P2+Q2(cid:27) (2.4)
(cid:12)(cid:12) (cid:12)(cid:12) exp 4β2e(P )2 AP −1σ2+4C22(P−1)α (k)N(P,k) AL +1
·  2  A 1 T  P +Q 
 − Xk=1 (cid:18) 2 2 (cid:19)
 
whereQ ,P suchthatQ P andP +Q <AL aswellas 
2 2 + 2 2 2 2
∈N ≤
A 1 logQ
β − andf :=2 2 .
≤ 4eCP (AP 1) logA
2 − (cid:24) (cid:25)
ProofofTheorem2.2. We have to partition V′ suitably. We use the abbreviations V( ) := V(L, ,P) and T :=
· ·
AL/(P +Q ) aswellas,
2 2
e
(cid:6) (cid:7)
A(i):=V (i 1)(P +Q )+1 ... V (i 1)Q +iP
2 2 2 2
− ∪ ∪ −
B(i):=V(cid:0)(i 1)Q2+iP2+1 (cid:1) ... V (cid:0)i(P2+Q2) , (cid:1)
e − ∪ ∪ e
(cid:0) (cid:1) (cid:0) (cid:1)
fori=1,...,T.NotethattheA(i)eandB(i)aretheunionofthedisjoeintsetsV( )andthatsomeA(i)andB(i)might
·
beempty.Furthermore,wedefine
V′ := T A(i)andV′ := T B(i).e
1 ∪i=1 2 ∪i=1
Then,wehavewithChebychev’sinequalityandthewell-knownAM-GMinequalitythat
e−βε
Zv >ε e2βPv∈V1′Zv + e2βPv∈V2′Zv forβ >0.
P vX∈V′ !≤ 2 nEh i Eh io
Hence,itsufficestoconsiderthesum Z closer. Wewrite
v∈V1′ v
P
S(i):= Z andJ(i):= Z fori=1,...,T.
v v
v∈∪Xii=1A(i) v∈XA(i)
We computetheexpectationsoftherandomvariableseδS(i), forδ > 0sufficientlysmall. Notethatthedistancew.r.t.
d betweenv A(i)andv′ A(i′),i=i′,isatleast2 logQ /logA . SinceS(i)=S(i 1)+J(i),weinferfrom
G 2
∈ ∈ 6 ⌈ ⌉ −
Davydov’sinequalitygiveninPropositionA.1that
eδS(i) =Cov(eδS(i−1),eδJ(i))+ eδS(i−1) eδJ(i)
E E E
h i h i h i
10α (f)1/a exp(δS(i 1)) exp(δJ(i)) + eδS(i−1) eδJ(i) (2.5)
≤ T k − kbk k∞ E E
h i h i
forHo¨lderconjugatea,b 1andf :=2 logQ /logA . Furthermore,wehaveif δJ(i) 1/(2e)that
2
≥ ⌈ ⌉ | |≤
expδJ(i) 1+δJ(i)+δ2J(i)2.
≤
NowtherandomvariablesZ areessentiallyboundedbyC. Letβ (A 1)/(4eCP (AP 1))anddefineδ := 2β.
v 2
≤ − −
5
Then,wehave
AP 1 1
δJ(i) δCP − , hence, [δJ(i)] exp δ2 J(i)2 .
2
| |≤ A 1 ≤ 2e E ≤ E
−
(cid:0) (cid:2) (cid:3)(cid:1)
Note that in the subgraphinduced by the A(i) there are exactly N(P,k) pairsof nodes(v,w) with d (v,w) = k
G
1,...,2(P 1) ,whereN(P,k)isgiveninLemma2.1. Forthenexttwolinesweusetheinequality( n a )2 ∈
{ − } i=1 i ≤
n2 n a2forrealnumbersa ,i=1,...,n,n . Consequently,weget
i=1 i i ∈N P
P
J(i)2 = Z2 + [Z Z ]
E E v E v w
(cid:2) (cid:3) v∈XA(i) (cid:2) (cid:3) v,wX∈A(i),
v6=w
AP 1 2(P−1)
(P )2 − σ2+4C2 α (k)N(P,k) =:K
≤ 2  A 1 T 
 − Xk=1 
withDavydov’sinequalityfromPropositionA.1. 
Furthermore,we find with the Ho¨lder inequalitythat exp(δS(i 1)) exp(δS(i 1)) . Thus, equation(2.5)
k − k1 ≤ k − kb
canbeboundedby
[exp(δS(i))] 10α (f)1/a exp(δJ(i) +exp(δ2K) exp(δS(i 1)) . (2.6)
E ≤ T k k∞ k − kb
n o
Especially, for the case i = T successive iteration of (2.6) yieldsfor the choicea := T +1 and b = 1+1/T (as in
Valenzuela-Dom´ınguezandFranke(2005))
[exp(δS(T))] exp 10√eα (f)1/(T+1)(T 1)+δ2eKT .
T
E ≤ { − }
Next,sinceα (f) 1and1/(1+T) P2+Q2 ,wearriveat
T ≤ ≥ 2(P2+Q2)+AL
AL
exp 2β Z exp 10√eα (f)(P2+Q2)/(2(P2+Q2)+AL)
v T
E  ≤ P +Q
vX∈V1′ (cid:26) 2 2(cid:27)
  
AP 1 2(P−1) AL
exp 4β2e(P )2 − σ2+4C2 α (k)N(P,k) +1 .
·  2  A 1 T  P +Q 
 − kX=1 (cid:18) 2 2 (cid:19)
 
 
The computations for exp 2β Z are similar and one achieves the same bounds for this term. This
E v∈V2′ v
finishestheproof. h (cid:16) (cid:17)i
P
Wearenowinpositiontoderiveaconcentrationinequality. Weconsideraninfinitetreewhichgrowsatanexponential
rateAandwhichisendowedwitharandomfieldZ. WeassumethattherandomfieldZ onthetreeT isstrongmixing
suchthat
α (k)N(P,k) O PAP , (2.7)
T
∈
k∈
XN (cid:0) (cid:1)
where N(P,k) is defined in Lemma 2.1. We say that the mixingcoefficientsdecayat a super-exponential(orhyper-
exponential)rateifthereisapositiveincreasingfunctiongwithlim g(n)= suchthat
n→∞
∞
α (n) exp( ng(n)). (2.8)
T
≤ −
In this case, equation (2.7) follows from Lemma 2.1 with the bound N(P,k) O PAP+k/2 and the following
∈
concentrationinequalityistrue
(cid:0) (cid:1)
Theorem 2.3 (Concentration inequality for exponentially growing trees). Let T = (V,E) be a tree growing at an
6
exponentialrateAandletZ bearandomfieldonT asinTheorem2.2. Lettherandomfieldbestrongmixingw.r.t.the
graphmetricwithα-mixingcoefficientswhichfulfill(2.7),e.g.,themixingcoefficientsdecayatasuper-exponentialrate
asin(2.8). ConsiderthesubgraphwhichconsistsofthefirstLgenerationsofT forL
∈N
V = (j,k) V :0 j L 1, 1 k AL−1 .
L
∈ ≤ ≤ − ≤ ≤
(cid:8) (cid:9)
Thenthereareconstantsc ,c suchthatforallL andε>0
1 2 +
∈R ∈N
1 L
Z >ε c exp c ε .
v 1 2
P |VL|(cid:12)(cid:12)vX∈VL (cid:12)(cid:12) !≤ (cid:26)− logL(cid:27)
(cid:12) (cid:12)
(cid:12) (cid:12)
Thismeanstheprobabilitydecaysasymptot(cid:12)icallyat(cid:12)aratewhichisapproximatelylinearinthesizeofthesampleVL.
ProofofTheorem2.3. LetP := Lη forsomeη (0,1). WepartitionV inthefollowingway: firstwedefinethe
1 L
⌊ ⌋ ∈
wedgewhichconsistofthefirstL P generations
1
−
W := (j,k) V : 0 j <L P .
L L 1
{ ∈ ≤ − }
TheremainingP generationsarecollectedin
1
U := (j,k) V : L P j <L .
L L 1
{ ∈ − ≤ }
The sums which correspond to these partitioning are S := Z and S := Z . Then we split the
L v∈WL v L v∈UL v
probabilityasfollows,
P P
e
ε ε
Z >ε V S > V + S > V (2.9)
v L L L L L
P (cid:12) (cid:12) | |!≤P | | 2| | P | | 2| |
(cid:12)vX∈VL (cid:12) (cid:16) (cid:17) (cid:16) (cid:17)
(cid:12) (cid:12) e
(cid:12) (cid:12)
Thefirstprobabilityin(2.9)i(cid:12)snegligib(cid:12)lebecausewefind
ε AL−P1 1 εAL 1
S > V 2C − > − .
L L
P | | 2| | ≤ 1 A 1 2 A 1
(cid:16) (cid:17) (cid:26) − − (cid:27)
e
Thus,wecanfocusonthesecondprobabilityin(2.9). WeuseTheorem2.2. Wemakethefollowingdefinitions
AL−P1
P :=Q := D logL , forasufficientlylargeconstantD
2 2 +
L P ∈R
(cid:22) − 1 (cid:23)
A 1 logP
β := − andf :=2 2 .
4eCP (AP1 1) logA
2 − (cid:24) (cid:25)
Consider the exponentof the first factor given in (2.4): one findsthat there is a constantc which doesneither
+
∈ R
dependonLnoronεnorontheZ suchthat
v
ε L
exp β V exp cε . (2.10)
L
−2 | | ≤ − logL
n o (cid:26) (cid:27)
Thesecondfactorin(2.4)isgivenby
exp B√eα (f)2P2/(4P2+AL−P1)AL−P1 . (2.11)
T
2P
(cid:26) 2 (cid:27)
Wecanderivethefollowingboundforthemixingcoefficientandtheexponentinsidetheexp-functionin(2.11)
α (f)2P2/(4P2+AL−P1) exp D logL 1+ log(DlogL/(L−P1)) .
T
≤ −5 (L P )logA
(cid:26) (cid:18) − 1 (cid:19)(cid:27)
7
Consider the second factor inside the exp-function in (2.11), it is AL−P1/P (L P )/logL. In particular, the
2 1
≤ −
secondfactorin(2.11)isuniformlyboundedforallL ifDissufficientlylarge.Considerthethirdfactorin(2.4).
+
∈N
Sincethemixingcoefficientsdecaysufficientlyfast,wecanderivethefollowinginequality
AP1 1 2(P1−1) AL−P1 P (L P )
exp 4eβ2(P )2 − σ2+4C2 N(P ,k)α (k) +1 exp c 1 − 1 ,
 2  A 1 1 T  2P ≤ AP1logL
 − Xk=1 (cid:18) 2 (cid:19) (cid:26) (cid:27)
 
forasuitableconstantc . Inparticular,thisexpressionisuniformlyboundedoverallL . Allinall, we have
∈ R ∈ N
shownthatthereareconstantsc ,c suchthatforthesecondprobabilityin(2.9)isboundedas
1 2 +
∈R
ε L
S > V c c ε ,
L L 1 2
P | | 2| | ≤ P − logL
(cid:16) (cid:17) (cid:18) (cid:19)
where,theasymptoticspeedisdeterminedby(2.10). Thiscompletestheproof.
Theprevioustheoremcanbeappliedtoexponentiallygrowinggraphsaswell,wehavetheusefulcorollary:
Corollary2.4(Concentrationinequalityforexponentiallygrowinggraphs). LetG = (V,E′ E˜)beagraphgrowing
∪
atanexponentialrateAendowedwitharandomfieldZ asinTheorem2.3. Thenthereareconstantsc ,c such
1 2 +
∈R
thatforallL andε>0
∈N
1 L
Z >ε c exp c ε .
v 1 2
P |VL|(cid:12)(cid:12)vX∈VL (cid:12)(cid:12) !≤ (cid:26)− logL(cid:27)
(cid:12) (cid:12)
(cid:12) (cid:12)
ProofofCorollary2.4. We only need to sh(cid:12)ow that(cid:12)the mixing conditions for the tree T = (V,E′) are fulfilled. The
conditionthatS :=sup d (v,w) (v,w) E˜ < impliesthat
T
{ | ∈ } ∞
d (v,w)
T
1 d (v,w) d (v,w).
G T
∨ S ≤ ≤
Inparticular,themixingratesw.r.t.thetreeandthewholegraphstructuresatisfyasymptoticallytheinequalityrelations
α ( S ) α α . Thus,wecanconcludethestatementfromTheorem2.3.
T G T
⌈ ·⌉ ≤ ≤
A Appendix
PropositionA.1(Davydov(1973)). Let(Ω,A, )beaprobabilityspaceandletG,H Abesub-σ-algebras.Denote
P ⊆
byα:=sup (A B) (A) (B) : A G,B G theα-mixingcoefficientbetweenGandH. Letp,q,r 1be
{|P ∩ −P P | ∈ ∈ } ≥
Ho¨lderconjugate.Letξ(resp. η)beinLp( )andG-measurable(resp. inLq( )andH-measurable).Then
P P
Cov(ξ,η) 10α1/r ξ η
| |≤ k kLp( )k kLq( )
P P
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