Nonparametric recursive density estimation for spatial data C.R. Acad. Sci. Paris, Ser.I

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Nonparametric recursive density estimation for spatial data

Estimation nonparamétrique récursive de la densité pour données spatiales

Aboubacar Amiri

^a

, Sophie Dabo-Niang

^a

^,

^b

, Mohamed Yahaya

^a

^,

^c

aUniversitéLille3,LaboratoireLEMCNRS9221,France bINRIA-MODALTeam,France

cFST,UniversitédesComores,Comoros

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received26February2014

Acceptedafterrevision18October2015 Availableonline23December2015 PresentedbytheEditorialBoard

Thispaper dealswith non-parametricdensityestimationfor spatialdata. Westudythe asymptoticpropertiesofanewrecursiveversionoftheParzen–Rozenblattestimator.The meansquareerrorandanalmostsureconvergenceresultwithrateofsuchestimatorare derived.

©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan} openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

r é s um é

Cepapiertraitedel’estimationdeladensitéspatialedanslecasrécursif.Nousétudionsles propiétésasymptotiquesd’unenouvelleversiondel’estimateurdeParzen–Rozenblatt.Nous établissonslesconvergencesenmoyennequadratiqueetpresquesûredecetestimateur ; desvitessesdeconvergencesontdonnées.

©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.Thisisan} openaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

Weconsiderrecursivekerneldensityestimationforspatiallydependentdata.Spatialdensityestimationisaninteresting andcrucialprobleminstatisticalinferenceforanumberofapplications,wheretheconsideredvariableshavespatialdepen- dence.Spatialdataaremodeledasfiniterealizationsofrandomfieldscollectedfromdifferentspatiallocationsontheearth andtheyare widelyusedina varietyoffields,includingsoilscience,geology,oceanography, econometrics,epidemiology, environmentalscience, forestry, andmanyothers,seeCressie [9],Chiles andDelfiner [7].Althoughpotential applications ofnonparametric spatialmodels areabundant, littletheoretical workhas beendevotedtononparametric spacemodeling comparedtotheparametriccase.

Thiswork concernsa nonparametricestimationof theprobability densityfunctionfromdependentspatial data,using a recursive kernel approach. For an overview on results and applications considering independent ordependent spatial data fordensityestimation by classical kernelmethod, we highlight theworks by Biau andCadre[3],Carbon et al.[5], Dabo-NiangandYao[10],Tran[19].Themethodandtheresultsobtainedheregeneralizethepreviousworksintherecursive framework.

E-mailaddresses:[email protected](A. Amiri),[email protected](S. Dabo-Niang),[email protected](M. Yahaya).

http://dx.doi.org/10.1016/j.crma.2015.10.010

1631-073X/©^{2015PublishedbyElsevierMassonSASonbehalfofAcadémiedessciences.ThisisanopenaccessarticleundertheCCBY-NC-NDlicense} (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Withtheadvancesinhardwaretechnologyandthesophisticationofmoderncomputinganddataacquisitiontechniques, voluminousspatialdataarecollectedinvariousappliedareas.Newchallengesarisesinceinmanyapplicationsthevolumeof thedataissolargethatitmaybeimpossibletostorethemondeskandtheirmodelingrequiresmanytime.Nonparametric estimatorssuchasthekerneldensityestimatorofParzen–Rozenblattcanbeusedtoestimatethedensityfunction.However, theysufferfromaseriouscomputationaldrawbackinbigdatacontext.Infact,ifthedensityisestimatedbyanon-recursive estimator,thislastmustbecompletelyrecalculatedwhennewdataitemsarereceived.

In situations wheredata itemsarrivesequentially anda largeamount ofdata canbe generated ata rapidrate, a re- cursive updating algorithms providegreat help for the difficulties arising from the large volume of data. Then they are muchpreferredovernon-recursiveones,sincerecursiveestimatorscanbeupdatedfromtheirimmediatepastandthenew observation. Therefore,theupdatecanbecomputedinstantlyanddoesnotrequireextensivestorageofdata.Thisarrange- ment isparticularto recursivemethodsandiscalledtheonlineorreal-timeupdatingproperty. Thisrecursive propertyis particularlyinterestingwhenthesampledataarecontinuallycapturedovertime.

Recursivekernelapproacheswere introducedandabundantlystudiedinthenon-spatialcase.Werefer,forinstance,to the pioneerworksby WolvertonandWagner [20],Deheuvels [12],Wegman andDavies [11], whohavepaid attentionto nonparametricrecursivedensityestimationfortimeprocesses.Therearesomerecentcontributionsonasymptoticproperties of recursive kernel densityfor dependent data. Amiri [1]generalized all the above-cited versions to a generalfamily of recursive estimators, usingdifferent valuesof a parameter

∈ [

⁰

,

1

]

^{associatedwitheach estimator. Forthe fixed values}

=

¹

/

2 and

=

^{1, the asymptotic normality under negative} association was previously treated by Liang and Baek [2], while themean-squareconvergenceandtheasymptoticnormality werestudiedby Masry[14] under

α

-mixingcondition.

Mezhoudetal.[15]haveextendedtheresultsobtainedbyAmiri[1]to

η

-dependenttimeprocess.

We extend the feature of time-varying sample recursive density estimate to the spatial case. Namely, we presenta recursive versionoftheclassical spatialkerneldensityestimatorandestablishsomeasymptoticresultsofsuchestimator.

Asfarasweknow,theinvestigationofrecursivekernelmethodsforspatialdataremainanopenproblem.

The restof the paper isorganized asfollows. In section 2, we present the spatial recursive density estimator, while Section 3 gives general assumptions. In section 4, we establishasymptotic convergence, namely mean square error and almost-sure convergence with the rate of our estimator. Finally Section 5 is devoted to some indications to prove the theoreticalresultspresentedinthisnote.

2. Therecursivespatialkerneldensityestimate

Let

(

Xi

)

_i∈NN

,

N

≥

^{1 be a measurable strictly stationary spatial process defined on a} probability space

(,

A

,

P

)

and valued inR^d,whered

≥

^{1.We assume that the X}i’shave thesame distributionasa random variable X,withunknown probability densityfunction p.Suppose that weobservethe process onaspatial set ofsitesIn ofcardinal n,whichis a finite subsetofapotentially observableregion D

⊂

R^N,whereR^{N isendowedwiththeuniformmetric.For}convenience, we treat the observationssites asan array that is In

=

s_j

,

j

=

¹

, . . . ,

n

⊂

N^{N. The estimator ofthe} probability density functionbasedon

(

X_i

,

i

∈

In

)

tobestudiedinthepresentworkisoftheform:

p_n

(

x0

) :=

¹ Sn,

i∈In

1 h^d_iK

X_i

−

^x0

h_i

,

x0

∈ R

^d

( ∈ [

0

,

1

]),

(1)

whereSn,

:=

i∈In

h^d_i⁽¹⁻⁾,h_iisabandwidthcorrespondingtotheobservationX_i.Byenumeratingthesites,onemayrewrite

p_n

(

x0

)

as

p_n

(

x₀

) :=

¹ Sn,

n j=¹

1 h^d_sj K

Xs_j

−

x0

hs_j

,

x₀

∈ R

^d

( ∈ [

⁰

,

1

] ),

(2)

whereSn,

:=

ⁿ

j=¹

h^d_s⁽_j¹⁻⁾.Thisrepresentationoftheestimatorpresentsagreatinterestfromthecomputationalpointofview.

Infact,ifXsn+1 isanewobservationoftheprocessatasitesn+¹ addedtoIn,theestimatorcanbeupdatedrecursivelyby thefollowingformula:

p_n+1

(

x₀

) =

^Sn,

S_n+1,p_n

(

x₀

) +

^K_n+1

X_sn+1

−

^x0

with K_i

(.) :=

¹ h^d_siS_i,K

.

h_si

.

(3)

In(3),p_n+1

(

x0

)

isthedensityestimatorbasedonthedomainIn

∪ {

^sn+¹

}

^{.Thisrecursiveformulaisusefulwhenthenumber} ofthespatialsitesincreasessequentiallyonspace.

Tosimplifythepresentation,theparameter

isthroughoutsupposedtobe equalto0 andthecorrespondingestimator willbelaternoted pn.

Basically,twoasymptoticmethodsoccurinthespatialliterature:increasingdomainandinfillasymptotics,see[9],p. 480.

The growthof thesample inincreasing domainasymptotics isa consequenceofan unbounded expansionof thesample

regionIn,whereasunderinfillasymptoticsthesampleregionisfixedandthegrowthofthesamplesizeisduetosampling thatisdenseintheregionD^{.Inwhatfollowsweconsidertheincreasingdomain}asymptoticsandforsimplicitythebivariate regularlattice(N

=

^{2),describedinthefollowing}assumption.

H2.1.D isaregularlatticeandIn

= {

ⁱ

= (

i₁

,

i₂

),

1

≤

ⁱj

≤

ⁿj

,

j

=

¹

,

2

}

^isrectangular.Inthiscase,n

=

ⁿ1

×

ⁿ2goestoinfinitymeans thatminn_{j j=1,2}goestoinfinity.Therefore,weusethelexicographicorderandrenumbertheobservationsasatriangulararrayinthe followingway.Theobservationsite

(

i

,

j

)

canbeindexedbyt

=

ⁿ2

(

i

−

¹

) +

^{j inthetriangulararraysetting,see[17].Inthiscase,the} newobservationsiteis

(

n₁

+

¹

,

1

)

,whichisindexedbyn

+

^{1inthearraysetting.}

Noticethattoobtainourasymptoticresultsintheirregularlatticecontext,onecanconsideradditionalassumptionson thenumberofspatialunitonaclosedballofDwiththefollowing:

H2.2.ThepotentialobservableregionD^{isanirregularlatticeandinfinitecountable.Allelementsof}D^{arelocatedatdistancesofat} least

δ >

1fromeachother.

Inthiscase(asin[8]),weconsiderasequenceoffiniteclosed,convexregions

{

^Ru

}

^{ofincreasingareaasu}

→ +∞

^andletthe samplesetconsistsoftheintersectionofoneoftheconvexregionsandD:In

=

D

⊂

^Ru.Wedonotspecifyhereanyorderandany otherassumptionontheconfigurationandgrowthofsamplesizeexcepttheassumptionofauniformlyincreasingareaof

{

^Ru

}

^inat leasttwonon-opposingdirectionstoincreasethesamplesizen (asu

→ +∞

^{),see[8].Thus,anew}observationcorrespondstoanynew sites_n+1ofD^{notalreadyincludedin}In.

3. Generalassumptions

Inordertoestablishtheasymptoticresults,thefollowingassumptionswillbeconsidered.Letusconsidertherepresen- tationoftherecursivedensityestimatorgivenin(2),with

=

^0.

H3.1.(a)hsn

↓

^0andnhdsn⁺²

→ ∞

^asn

→ ∞

^;(b)Forr

∈ ]−∞ ,

2

+

^d

]

^,Bn,r

:=

¹ n

n i=¹

hsi

h_sn r

→

^Br

>

0.

H3.2.K isaboundedsymmetricdensitysuchthat

R^d

u K

(

u

)

du

=

^0and

R^d

^u

^2K

(

u

)

du

< ∞

^.

H3.3.Thedensityp isboundedandtwicecontinuouslydifferentiableonR^d.

H3.4.Foranyi

,

j

∈ {

¹

, . . . ,

n

}

^suchthatsi

=

^sj,therandomvector

(

X_si

,

X_sj

)

hasadensity f_si,sjsuchthat:

sup

si=sj

gs_i,s_j

∞

< ∞ ,

where gs_i,s_j

=

^fs_i,s_j

−

^p

⊗

^p

,

i

,

j

=

¹

, . . . ,

n

.

H3.5.

(i) Thefield

(

X_si

)

1≤ⁱ≤ⁿis

α

-mixing:thereexistsafunction

φ :

R⁺

→

R^+with

φ (

t

)

^0ast

→ ∞

^{,suchthatwheneverE}

,

E

⊂

R² withfinitecardinalsCard

(

E

)

,Card

(

E

)

α σ (

E

) , σ

E

:=

^sup

A∈σ(E),B∈σ(E)

|P(

^A

∩

^B

) − P(

^A

) P(

^B

)| ≤ ψ

Card

(

E

),

Card

(

E

) φ

dist

(

E

,

E

) ,

where

σ (

E

) = {

^Xi

,

i

∈

^E

}

^and

σ

E

=

Xi

,

i

∈

^E

,dist

(

E

,

E

)

istheEuclideandistancebetweenE andEand

ψ( · )

isapositive symmetricfunctionnondecreasingineachvariable.

Thefunctions

φ

and

ψ

aresuchthat

φ (

i

) ≤

^{C i−θand}

ψ(

n

,

m

) ≤

^Cmin

(

n

,

m

)

. (ii) For

>

0,letun

=

N

i=¹

(

logni

)(

log logni

)

¹⁺andassumethat

nlogn^{θ−4N−θ2N}h

dθ θ−^4N

n u

2N 4N−θ

n

→ ∞,

^where

θ >

max

4N

,

^Nd

+

² 2

.

ThefirstconditionofH3.1isusualinnonparametricrecursiveestimation,whilethesecondoneiscrucialinourcalculus.

This last is particularly used in the recursive kernel estimation, for instance by Masry [14], Wegman and Davies [11], Yamato[21],SamantaandMugisha[16]andIsogai[13].Ifh_sn

=

^Ann^−q whereA_n

↓

^A

>

0

,

0

<

q

<

1,thentheassumption H 3.1 is satisfied for all q such that rq

<

1 and Br

=

¹

1

−

^{rq. Also the choice hsn}

=

^An

lnn n

q

,

0

<

q

<

¹

2

+

^{d satisfies} assumption H 3.1. Assumptions H 3.2 and H 3.3 are classical in nonparametric estimation and easily verified by many kernelssuchasEpanechnikov,Gaussiankernels.

AbouttheassumptionH3.5(i),basically,inthespatialliteratureitissupposedthat

φ (

i

)

tendstozerowithpolynomial rate, or

φ (

i

) ≤

^Cexp

(−

^si

),

for someC

,

s

>

0,i.e.

φ (

i

)

tendstozerowithexponentialrate.Ourresultscanbeprovedunder theabovecondition.Finallyletusmentionthatthedefinitionofun inH3.5leadsto

n∈N^Nnu¹n

< ∞

^(recallthatn

=

^n).

4. Consistencyresults

Theorem 1belowestablishestheasymptoticmeansquareerroroftheestimatorp_n viaitsvarianceandbias.

Theorem1.UndertheassumptionsH2.2 andH3.1–3.5,wehave:

nh^d_snVar

(

pn

(

x0

)) →

^p

(

x0

)

B⁻_d¹

R^d

K²

(

u

)

du (4)

and

h⁻_s²

n

(E(

^pn

(

x₀

)) −

^p

(

x₀

)) →

^B−_d^1Bd+²

R^d K

(

u

)

2

d j₁,j₂=¹

u_j1u_j2

∂

²p

(

x₀

)

∂

x_j1

∂

x_j2du (5)

asn

→ ∞

^,forallx0suchthatp

(

x0

) >

0.Inparticular,ifhsn

=

^{n4+d−1,wehavetheasymptoticbehavioraccordingtothemeansquare} error:

n^4+4d

E(

^pn

(

x₀

) −

^p

(

x₀

))

²

→

⎛

⎜ ⎝

R^d K

(

u

)

d j₁,j₂=¹

u_j1u_j2

∂

²p

(

x₀

)

∂

x_j1

∂

x_j2du

⎞

⎟ ⎠

2

+

⁴ 4

+

^dp

(

x₀

)

R^d

K²

(

u

)

du (6)

asn

→ ∞

^forallx0suchthatp

(

x0

) >

0.

Theorem 1isanextensionofthepreviousresultsobtainedinthetemporalcasebyAmiri[1]with

α

-mixingcondition.

Also, asimilarresultwereprovedbyMezhoudetal.[15] fortemporal

η

-dependenceprocess.Now,westatethefollowing theoremthatestablishesthealmostconvergenceofpn.

Theorem2.UndertheassumptionsH2.1,H3.1–3.5,wehave:

|

^pn

(

x₀

) −

^p

(

x₀

)| =

^O

logn

nh^d_n

,

a

.

s

.

(7)

5. Briefoutlineofproofs 5.1. Theorem 1

Forx₀

∈

R^{d,let I}1

=

¹ B_n²_,dn²h^2d_sn

n i=¹

Var Z_si

andI₂

=

¹ B²_n,dn²h^2d_sn

si=^sj

Cov Z_si

,

Z_sj

,

with Z_si

=

^K

Xs_i

−

^x0 hsi

. Ontheonehand,underH3.2withthehelpoftheToeplitzLemma(see[18]),wereadilyhave

nh^d_s

n

|

^I1

| →

^p

(

x₀

)

B⁻_d¹

R^d

K²

(

u

)

du asn

→ ∞

^.

Ontheotherhand,forx0

∈

R^{d,letussplit:I}2

=

^J1

+

^J2,where J1

=

¹

B²_n,dn²h^2d_sn

_si−sj≤cn

Cov

(

Zs_i

,

Zs_j

)

and J2

=

¹ B²_n,dn²h^2d_sn

cn<_si−sjCov

(

Zs_i

,

Zs_j

),

cn beingasequencetendingtoinfinityasntendstoinfinity.Ifd

≥

^3,then2d

>

d

+

^{2.Inthiscase,weconsider}

ζ ∈

R^such that N

θ −

¹

< ζ ≤

²

d,andusingthedecreaseof

(

h_sn

)

n,onemaywrite:

Cov

(

Z_si

,

Z_sj

) ≤

^hd_si^hds_j sup

si=sj

g_si,sj

∞

≤

^h

2ds_i

+

^h2ds_j

2 sup

si=sj

g_si,sj

∞

≤

^h

d(ζ+¹)

s_i h^d_s1^{(1−ζ )}

+

^hds^(ζ+_j ¹⁾h^d_s1^{(1−ζ )}

2 sup

si=sj

g_si,sj

∞

.

If1

≤

^d

≤

^3,take

ζ =

^{1 andmakeas}previously.AsIn isaregulardesign(seeforinstance[19]) Card

(

u

,

v

) ∈

I_n²

:

^u

−

^v

≤

^cn

=

^O

nc_n^N

.

Itfollowsthatnh^d_sn

|

^J1

| ≤

^Bn,d(ζ+1)

B²_n,d h^d_s1^{(1−ζ )}h^d_sn^ζc_n^Nsup si=^sj

gs_i,s_j

∞

=

^O h^d_sn^ζc^N_n

.

Turningto J₂,bytheBillingsleyinequality(see[4]),wehaveCov

(

Z_si

,

Z_sj

) ≤

⁴

φ (

s_i

−

^sj

)

^K

²_∞^.Consequently,weget nh^d_s

n

|

^J2

| ≤

⁴

^K

²_∞ B²_n,dh^d_sn

(θ −

¹

)

^c

1−θ

n

=

^O

c_n^1−θ h^d_sn

.

The choice ofcn

=

h⁻

d(ζ+1) N+θ−1 sn

leads to nh^d_sn

|

I2

| =

O

⎛

⎜⎝^h

d

(ζ (θ −

¹

) −

^N

)

N

+ θ −

¹

sn

⎞

⎟⎠

→

^{0 asn}

→ ∞

^{, aswell as}

ζ >

^N

θ −

^1,andthe

result(4)follows.Aboutthebiasterm,weapply theTaylorformulaandthedominatedconvergence,whichtogether with Toeplitz’lemma,leadstotheresult(5). 2

5.2. Theorem 2

LetussetG_n

(

x₀

) =

^pn

(

x₀

) −

E^pn

(

x₀

) =

n

i=¹

iwith

i

=

¹

Sn,0

Z_si

−

E^Zsi

.Next,leta

=

^an bean integerandsplitthe randomvariables

i intoblocksasfollows:

U

(

1

,

n

,

j

,

x0

) =

(2j

k+1)an i_k=^2jkan+¹

k=¹,...,N

_i

;

^U

(

2

,

n

,

j

,

x0

) =

(2jk

+1)an i_k=^2jkan+¹ k=¹,...,N−¹

2(jN

+1)an iN=(2jN+¹)an+¹

_i

;

U

(

3

,

n

,

j

,

x0

) =

(2j

k+1)an i_k=^2jkan+¹ k=¹,...,N−²

2(j_N

₋₁+¹)a_n iN−1=(2jN−1+¹)an+¹

(2jN

+1)an iN=^2jNan+¹

_i

;

U

(

4

,

n

,

j

,

x0

) =

(2j

k+1)an i_k=2j_kan+1 k=¹,...,N−²

2(j_N

₋₁+¹)a_n iN−1=(2jN−1+1)an+1

2(jN

+1)an iN=(2jN+1)an+1

_i

,

(8)

andsoon.FinallynotethatU

(

2^N−1

,

n

,

j

,

x0

) =

(2jk+1)an ik=^2jkan+¹ k=¹,...,N−¹

2(jN+1)an iN=^2jNan+¹

_i

;

^U

(

2^N

,

n

,

j

,

x0

) =

(2jk+1)an ik=^2jkan+¹

k=¹,...,N

_i.

Foreachinteger1

≤

ⁱ

≤

^{2N,define E}

(

n

,

i

,

x₀

) =

r_k−¹ j_k=⁰ k=1,...,N

U

(

i

,

n

,

j

,

x₀

)

,thenonemaywriteG_n

(

x₀

) =

₂^N+1

i=¹ E

(

n

,

i

,

x₀

)

.Without

lossofgenerality,wewillshowtheresultfori

=

^1.

Because K is bounded, we get U

(

1

,

n

,

j

,

x₀

) ≤

^aNn_Bn,d^K∞_nh¹d n

. Consider the sequences

λ

n

=

^nhdnlogn1/2!

, and r_i

=

n_i

/(

2a_n

),

i

=

¹

, . . . ,

N.Wededucethat,fornlargeenough,

λ

n

|

^U

(

1

,

n

,

j

,

x0

) | =

^O

⎛

⎜ ⎝ "

^logn nh^d_n

⎞

⎟ ⎠ .

(9)

Enumerate the randomvariables U

(

1

,

n

,

j

,

x

)

in E

(

n

,

1

,

x

)

inan arbitrary mannerand refer tothem as U

ˆ

1

, . . . ,

U

ˆ

M, M

=

r1

...

rN.ByMarkov’sinequalityandLemma4.5of[6],thereexistr.v.’sU

˜

1

, . . . ,

U

˜

M independentofU

ˆ

1

, . . . ,

U

ˆ

M andsuchthat

ˆ

U_i

= ˆ

^Ui

(

x₀

)

hasthesamedistributionasU

˜

_i

= ˜

^Ui

(

x₀

)

and:

P

U

ˆ

_i

− ˜

^Ui

> ≤

¹⁸

⎛

⎜ ⎝

U

ˆ

_i

∞

⎞

⎟ ⎠ ψ (

n

,

a^N_n

)φ (

a_n

).

(10)

Let _n

= η

logn

nh^d_n ,where

η

isapositivenumber.Obviously

P(|

E

(

n

,

1

,

x0

)| >

n

) ≤ P

⎛

⎝

^M

j=1

_U

˜

_i

>

n

2

⎞

⎠ + P

⎛

⎝

^M

j=1

_U

ˆ

_i

− ˜

Ui

>

n

2

⎞

⎠ .

(11)

UsingtheindependenceoftheU

˜

i’sandapplyingBernsteininequality,weget

P

⎛

⎝

^M

j=1

U

˜

i

>

n

2

⎞

⎠ ≤

2 exp

− λ

n

n

2

+ λ

_n² 4

M i=1

E

U

ˆ

²_i

(

x0

) ≤

2 exp

− η

logn 2

+ λ

²_n

4

M

i=1

E

U

ˆ

_i²

(

x0

) .

Clearly,

λ

²_n 4

M i=¹

E

ˆ

U_i²

(

x₀

)

≤ λ

²_n

4Var

(

p_n

(

x₀

)) ≤

^log

(

n

)

4 nh^d_nVar

(

p_n

(

x₀

))

.

From(4),wededucethatfornlargeenough,nh^d_nVar

(

pn

(

x0

)) ≤

^p

(

x0

)

B_d⁻¹

R^d

K²

(

u

)

du.Itfollowsthat

λ

²_n 4

M i=¹

E

U

ˆ

²_i

(

x0

)

≤

^log

(

n

)

p

(

x0

)

^B

−¹ d

4

R^d

K²

(

u

)

du

.

HenceP

⎛

⎝^M

j=¹

U

˜

_i

>

n 2

⎞

⎠

≤

^n−δ,where

δ >

1,for

η

largeenough.Concerningthesecondtermintheright-hand-sideof(11),

from(10)onehasU

ˆ

_i

(

x₀

)

∞

≤

a^N_n

^K

_∞ nh^d_n

.Weget,byMarkov’sinequality:

P

_M

i=¹

U

ˆ

_i

− ˜

^Ui

>

_n

2

≤

^{C M}

a_n^N

^K

_∞ nh^d_n

_n⁻¹

ψ (

n

,

a_n^N

)φ (

an

),

thenP _M

i=¹

U

ˆ

_i

− ˜

^Ui

>

n 2

≤

^C

1 h^d_n

_n⁻¹

ψ(

n

,

a_n^N

) (φ (

an

)) ≤

^C 1

h_n^d

_n⁻¹a_n^Na⁻_n^θ.

Letan

=

logn nh^d_n

₋1/(2N) ,thenP

_M

i=1

U

ˆ

i

− ˜

^Ui

>

n

2

≤

^{Cn2N−θ2N lognθ−2N2N h}n^−dθ2N

=

^C

β

n. By hypothesis, we have nu_n

β

n

=

nlogn^{θ−2N4N−θ}h

θ−4Ndθ

n u

4N−θ2N n

^4N−θ_2N

→

^{0, then Theorem 2 follows by the} Borel–Cantelli lemma. 2

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