asymptotic normality of recursive density estimates under some dependence assumptions

Abstract. Let fXn; n � 1g be a strictly stationary sequence of negativelyassociated random variables with the marginal probability density functionf ðxÞ, the recursive kernel estimate of f ðxÞ is defined by

fnðxÞ ¼1

n

Xn

j¼1h�1j Kðx� Xj

hjÞ;

where hn is a sequence of positive bandwidths tending to 0, as n!1, Kð�Þ isa univariate kernel function. In this note, we discuss the point asymptoticnormality for fnðxÞ.

Key words: Negatively associated random variables; Recursive kernel esti-mate; Asymptotic normality

AMS 1991 Subject Classification: 62G05

1 Introduction

In many stochastic models, the assumption that random variables are inde-pendent is not plausible. Increases in some random variables are often relatedto decreases in other random variables so an assumption of negative depen-dence is more appropriate than an assumption of independence. Lehmann(1966) investigated various conceptions of positive and negative dependencein the bivariate case. Strong definitions of bivariate positive and negativedependence were introduced by Esary and Proschan (1972). These were laterdeveloped by Alam and Saxena (1981), Block, Savits and Shaked (1982), theirdefinition is:

A finite family of random variables fXi; 1 � i � ng is said to be negativelyassociated (NA) if for every pair of disjoint subsets A and B of f1; 2; � � � ; ng,

Metrika (2004) 60: 155–166DOI 10.1007/s001840300302

Asymptotic normality of recursive density estimatesunder some dependence assumptions

Han-Ying Liang1 and Jong-Il Baek2

1 Department of Applied Mathematics,Tongji University, Shanghai 200092, P. R. China(E-mail: [email protected])

2 School of Mathematics & Informational Statistics and Institute of Basic Natural Science,Wonkwang University, Ik-San 570-749, South Korea (E-mail: [email protected])

Covðf1ðXi; i 2 AÞ; f2ðXj; j 2 BÞÞ � 0

whenever f1 and f2 are coordinatewise increasing and such that the covari-ance exists. An infinite family of random variables is NA if every finitesubfamily is NA.

This definition is carefully studied by Joag-Dev and Proschan (1983). Theycompared it with other concepts of negative dependence, and justified theclaim that NA possesses certain advantages over competing notions of neg-ative dependence. They also derived, as a by-product of their main results,that many well-known multivariate distributions are NA. Because of its wideapplications in multivariate statistical analysis and systems reliability, thenotion of NA has received considerable attention recently. For convergenceresults, we refer to Joag-Dev and Proschan (1983) for fundamental properties,Shao and Su (1999) for law of the iterated logarithm, Liang (2000) forcomplete convergence and Roussas (1994) for the central limit theorem ofrandom fields. Bozorgnia et al. (1993) also derives a wealth of resultsregarding limiting theorems for NA random variables. Asymptotic propertiesof estimates related to NA samples have also been studied extensively. Caiand Roussas (1998) studied uniformly strong consistency, convergence ratesand asymptotic distribution of Kaplan-Meier estimator of a distributedfunction with random censored failure times, Cai and Roussas (1999) gaveBerry-Esseen bounds for smooth estimates of a distribution function, Rous-sas (2001) investigated consistency of the kernel estimate of a probabilitydensity function and Amini and Bozorgnia (2000) dealt with the consistencyand complete convergence of sample quantiles.

Nonparametric estimation of a probability density is an interestingproblem in statistical inference and has an important role in communicationtheory and pattern recognition. The purpose of this paper is to investigaterecursive density estimators when the observations are NA random samples.

Throughout this paper, let fXn; n � 1g be a strictly stationary sequence ofnegatively associated random variables with the marginal probability densityfunction (p.d.f.) f ðxÞ, the recursive kernel estimate of f ðxÞ is defined by

fnðxÞ ¼1

n

Xn

j¼1h�1j Kðx� Xj

hjÞ;

which was introduced by Wolverton and Wagner (1969) and apparentlyindependently by Yamato (1971). Note that fnðxÞ can be computed recur-sively by

fnðxÞ ¼n� 1

nfn�1ðxÞ þ ðnhnÞ�1Kð

x� Xn

hnÞ:

This property is particularly useful in large sample size since fnðxÞ can beeasily updated with each additional observation. Here hn is a sequence ofpositive bandwidths tending to 0, as n!1, Kð�Þ is a univariate kernel.

In the independent case, fnðxÞ has been thoroughly examined in Wegmanand Davies (1979). In the dependent case, quadratic mean convergence andasymptotic normality of these recursive estimators have been obtained byMasry (1986) under various assumptions on the dependence of Xi. Strongpointwise consistency of fnðxÞ has been proved by Gyorfi (1981). Takahata(1980) and Masry and Gyorfi (1987) obtained sharp almost sure rates of fnðxÞ

156 H.-Y. Liang and J.-Il. Baek

to f ðxÞ for the class of asymptotically uncorrelated processes, the definition ofwhich can be found in Masry and Gyorfi (1987). Masry (1987) establishedsharp rates of almost sure convergence of fnðxÞ to f ðxÞ for vector-valuedstationary strong mixing processes under weak assumptions on the strongmixing condition, these rates were improved by Tran (1989). Tran (1990)studied the uniform convergence and asymptotic normality of fnðxÞ undersome dependent assumption defined in terms of joint densities.

In addition, a closely related estimator is the Rosenblatt-Parzen kernelestimate of f ðxÞ

f nðxÞ ¼1

nhn

Xn

j¼1Kðx� Xj

hnÞ;

which has been extensively studied by many authors, such as Wolverton andWagner (1969), Roussas (2000,2001), Bosq, Merlevede and Peligrad (1999)and Lu (2001).

In this note, we shall discuss the point asymptotic normality for fnðxÞ. Themethods of proof are closely related to those of Masry (1986), Tran (1990)and Roussas (2000). Here, unlike mixing cases, the negatively associatedrandom variable X1;X2; � � � ;Xn are subject to the transformationKðx� Xj=hjÞ; j ¼ 1; 2; � � � ; n, losing in this process the negatively associatedproperty, i.e. the kernel weights Kðx� Xj=hjÞ are not necessarily negativelyassociated.

In the sequel, let C denote a positive constant, CðxÞ and Cðx; yÞ denotepositive constants depending on x and x; y, respectively, whose values areunimportant and may vary at different place. The set cðf Þ, < and N denotethe continuity points of the function f , the real numbers and the naturalnumbers, respectively; suppðf Þ ¼ fx 2 <; f ðxÞ > 0g.

Now, we shall give some assumptions:

(A1) If f ðx; y; kÞ is the joint p.d.f. of the random variable Xj and Xjþk, thensupx;y jf ðx; y; kÞ � f ðxÞf ðyÞj � M0 for k � 1.

(A2) (i) The kernel function K satisfies

K 2 L1;

Z

<KðuÞdu ¼ 1; sup

x2<ð1þ jxjÞjKðxÞj <1:

(ii) The derivative ðd=duÞKðuÞ ¼ K 0ðuÞ exists for all u 2 < and isbounded jK 0ðuÞj � B for u 2 <.

(A3) 0 < hn # 0; hnPn

j¼1 h�1j =n! h ð0 < h <1Þ.(A4) Let 0 < p ¼ pn < n; 0 < q ¼ qn < n be integers tending to1 along with

n, and let hn > 0 be bandwidths and k ¼ kn ¼ ½ npþq� ! 1 so that

kðp þ qÞ=n! 1 and (i) pnknn ! 1, (ii) pnhn ! 0 and p2

n=nhn ! 0, (iii)1h3n

P1j¼qnjCovðX1;Xjþ1Þj ! 0:

Remark 1.1 (a) The assumption (A1) was used by many authors, (A2)(i) and(A3) are similar to that used by Masry (1986); in addition, (A2)(i)implies that K is bounded and K 2 L2 from K 2 L1.

(b) Since qnkn=n ¼ ðpn þ qnÞkn=n� pnkn=n; (A4)(i) implies qnkn=n! 0.Also,qn=pn ¼ ðqnkn=nÞ=ðpnkn=nÞ ! 0; so that qn < pn, eventually.

Asymptotic normality of recursive density estimates 157

(c) The first of assumptions (A4)(ii) and (b) imply that qnhn ! 0, which alsoimplies hn ! 0: The second of the assumptions in (A4)(ii) impliesnhn !1:

(d) Assumptions (A4)(i)(ii) are easily satisfied, if pn and qn are chosen asfollows: With hn ! 0, let pn � h�d1

n ; qn � h�d2n ð0 < d2 < d1 < 1Þ, where

xn � ynmeans that, as n!1; xn=yn tends to a constant. It is easily seenthat kn � nhd1

n , so that (A4)(i)(ii) are satisfied, provided nh1þ2d1n !1:

Also, for Assumption (A4)(iii), let jCovðX1;Xjþ1Þj ¼ Ckj ð0 < k < 1Þ, then

h�3n

X1

j¼qn

jCovðX1;Xjþ1Þj � h�3n kqn � h�3n = exp½ð� log kÞh�d2n � ! 0

as n!1. Thus, (A4) is satisfied under the condition that nh1þ2d1n !1:

Next, let jCovðX1;Xjþ1Þj ¼ Cj�a ða > 1Þ, then

h�3n

X1

j¼qn

jCovðX1;Xjþ1Þj � h�3þða�1Þd2n ! 0;

provided, a > 1þ 3=d2.Thus, (A4) is satisfied for a > 1þ 3=d2 andnh1þ2d1

n !1.(e) The assumptions in Assumptions (A4) can be seen as purely technical for

proving our main result. But (A4)(iii) can also be seen as a ‘‘weakdependence’’ assumption, which is a further restriction for NA variables.

Our main results are as follows:

Theorem 1.1 Assume that (A1)-(A4) hold true.

(1) For x 2suppðf Þ \ cðf Þ,lim

n!1nhnVarðfnðxÞÞ ¼ r2ðxÞ :¼ hf ðxÞ

Z

<K2ðuÞdu: ð1:1Þ

ZnðxÞ ¼ffiffiffiffiffiffiffinhn

p½fnðxÞ � EfnðxÞ�

D�!n!1Nð0; r2ðxÞÞ :¼ ZðxÞ: ð1:2Þ

(2) In addition, if x1; x2; � � � ; xd are distinct points of suppðf Þ \ cðf Þ, thenðZnðx1Þ; Znðx2Þ; � � � ; ZnðxdÞÞ

D�!n!1ðZðx1Þ; Zðx2Þ; � � � ; ZðxdÞÞ; ð1:3Þ

where Zðx1Þ; Zðx2Þ; � � � ; ZðxdÞ are independent.

By applying the Toeplitz lemma (see Hall and Heyde (1980), p. 31 or Masry(1986)) and the Taylor expansion, we can obtain that:

Lemma 1.1 Suppose that (A2)(i) holds.

(1) For x 2 cðf Þ, we have limn!1 EfnðxÞ ¼ f ðxÞ:(2) Assume that the second-order derivative f 00 of f exists and is continuous and

bounded, and that Kð�Þ satisfiesZ

<uKðuÞdu ¼ 0;

Z

<u2KðuÞdu <1;


fhn; n � 1g satisfyP1

n¼1 h2n ¼ 1: Then

limn!1ð1n

Xn

j¼1h2

j Þ�1½EfnðxÞ � f ðxÞ� ¼ 1

2f 00ðxÞ

Z

<u2KðuÞdu:

Corollary 1.1 Denote by Z�nðxÞ ¼ffiffiffiffiffiffiffinhnp

½fnðxÞ � f ðxÞ�. Suppose that all theassumptions of Theorem 1.1 are fulfilled, and that the conditions of Lemma 1.1hold and ðn�1hnÞ1=2

Pnj¼1 h2

j ! 0. If Z�n replaces Zn, then the results in Theorem1.1 are still true.

2 Proofs of Main Result

Lemma 2.1 (Masry (1986)) Assume that K satisfies (A2)(i). If g 2 L1, then forx 2 cðgÞ,

limh!0

h�1Z

<Kðx� u

hÞgðuÞdu ¼ gðxÞ

and

limh!0

h�1Z

<Kðx� u

hÞKðy � u

hÞgðuÞdu ¼ gðxÞ

R< K2ðuÞdu; x ¼ y;0; x 6¼ y:

�

Lemma 2.2 (Cai and Roussas (1999) Let A and B be disjoint subsets of N,andlet fXj; j 2 A [ Bg be NA. Assume that f : <#A ! < and g : <#B ! < arepartially differentiable with bounded partial derivatives, denote byk@f =@tik1stands for the supnorm. Let q : < ! < be a bounded differentiablefunction with bounded derivative. Then

jCovff ðXi; i 2 AÞ; gðXj; j 2 BÞgj �X

i2A

X

j2B

k @f@tik1 � k

@g@tjk1½�CovðXi;XjÞ�;

jCovfY

i2A

qðXiÞ;Y

j2A

qðXjÞgj � kqk#Aþ#B�21 kq0k21

X

i2A

X

j2B

jCovðXi;XjÞj:

Proof of Theorem 1.1 We only prove (1.3), the proof of (1.1) and (1.2) isanalogous. By the Cramer-Wold method, we only have to verify that for anynon-zero real number a1; a2; � � � ; ad ,

Xd

i¼1aiZnðxiÞ �!

D Xd

i¼1aiZðxiÞ: ð2:4Þ

Without loss of generality, we shall prove (2.4) for d ¼ 2, i.e.

aZnðxÞ þ bZnðyÞ �!D

aZðxÞ þ bZðyÞ; ð2:5Þwhere a ¼ a1; b ¼ a2; x ¼ x1; y ¼ x2:

Denote by Kðx�Xj

hjÞ ¼ Kðx�Xj

hjÞ � EKðx�Xj

hjÞ. Hence, (2.5) becomes into


Xn

j¼1

ffiffiffiffiffihn

n

rh�1j ½aKðx� Xj

hjÞ þ bKðy � Xj

hjÞ� �!D Nð0; a2r2ðxÞ þ b2r2ðyÞÞ: ð2:6Þ

For m ¼ 1; 2; � � � ; k, split the set f1; 2; � � � ; ng into k(large) p-blocks, Im, andk(small) q-blocks, Jm, as follows:

Im ¼ fi : i ¼ ðm� 1Þðp þ qÞ þ 1; � � � ; ðm� 1Þðp þ qÞ þ pg;Jm ¼ fj : j ¼ ðm� 1Þðp þ qÞ þ p þ 1; � � � ;mðp þ qÞg;

the remaining points form the set fl : kðp þ qÞ þ 1 � l � ng (which may be ;).Put

nnj ¼ffiffiffiffiffihn

n

rh�1j ½aK ðx� Xj

hjÞ þ bKðy � Xj

hjÞ�

and for m ¼ 1; 2; � � � ; k, set

ynm ¼X

i2Im

nni; y0nm ¼X

j2Jm

nnj; y00nk ¼Xn

l¼kðpþqÞþ1nnl:

Also set

Sn ¼Xn

j¼1nnj; Tn ¼

Xk

m¼1ynm; T 0n ¼

Xk

m¼1y0nm; T 00n ¼ y00nk :

So that

Sn ¼ Tn þ T 0n þ T 00n :

and convergence (2.6) will be established by showing that

Tn �!D

Nð0; a2r2ðxÞ þ b2r2ðyÞÞ: ð2:7Þand

EðT 0nÞ2 þ EðT 00n Þ

2 ! 0: ð2:8ÞTo prove (2.7) and (2.8), we shall divide them into the following four lemmas.

Lemma 2.3 Under the assumptions of Theorem 1.1, we have VarðT 0nÞ ! 0:

Proof. Note that

VarðT 0nÞ ¼Xk

m¼1Varðy0nmÞ þ 2

X

1�i<j�k

Covðy0ni; y0njÞ: ð2:9Þ

While

Varðy0nmÞ¼XmðpþqÞ

i¼ðm�1ÞðpþqÞþpþ1VarðnniÞþ2

X

ðm�1ÞðpþqÞþpþ1�i<j�mðpþqÞCovðnni;nnjÞ

:¼I1ðmÞþ I2ðmÞ:Note that h�1j VarðKðx�Xj

hjÞÞ ! f ðxÞ

R< K2ðuÞdu; j!1; which shows that for

all j � 1,


h�1j VarðKðx� Xj

hjÞÞ � CðxÞ: ð2:10Þ

By Lemma 2.1 , we get

jh�2j CovðKðx� Xj

hjÞ;Kðy � Xj

hjÞÞj

¼ jh�2j

ZKðx� u

hjÞKðy � u

hjÞf ðuÞdu

� h�2j

ZKðx� u

hjÞf ðuÞdu �

ZKðy � u

hjÞf ðuÞduj

� Ch�1j þ Cðx; yÞ: ð2:11Þ

Applying (2.10) and (2.11) and the monotonocity of hj, we get

I1ðmÞ ¼XmðpþqÞ

j¼ðm�1ÞðpþqÞþpþ1

hn

nh�2j ½a2VarðKðx� Xj

hjÞÞ þ b2VarðKðy � Xj

hjÞÞ

þ 2abCovðKðx� Xj

hjÞ;Kðy � Xj

hjÞÞ�

� a2qCðxÞn

þ b2qCðyÞn

þ Cqn� 2abþ 2abqhn

n� Cðx; yÞ: ð2:12Þ

Note that, for i < j, by using (A1), we have

supx;yjCovðKðx�Xi

hiÞ;Kðy�Xj

hjÞÞj

¼ hihj supx;yjZ

<2

KðsÞKðtÞ½f ðx� shi;y� thj;j� iÞÞ� f ðx� shiÞf ðy� thjÞdsdt�j

�M0hihj;

and hence

I2ðmÞ ¼2hn

n

X

ðm�1ÞðpþqÞþpþ1�i<j�mðpþqÞh�1i h�1j fa2CovðKðx� Xi

hiÞ;Kðx� Xj

hjÞÞ

þ b2CovðKðy � Xi

hiÞ;Kðy � Xj

hjÞÞ þ ab½CovðKðx� Xi

hiÞ;Kðy � Xj

hjÞÞ

þ CovðKðy � Xi

hiÞ;Kðx� Xj

hjÞÞ�g � 2M0ða2 þ b2 þ 2abÞ q

2hn

n:

ð2:13ÞTherefore, by (2.12), (2.13) and applying (A4) we get

Xk

m¼1Varðy0nmÞ �ða2CðxÞ þ b2CðyÞ þ 2abCÞ kq

nþ 2abCðx; yÞ � kq

nhn

þ 2M0ða2 þ b2 þ 2abÞ kq2hn

n! 0: ð2:14Þ

By Lemma 2.2 , (A2)(ii) and noticing i 6¼ j, we get


jCovðy0ni;y0njÞj�

hn

n

XiðpþqÞ

s¼ði�1ÞðpþqÞþpþ1

XjðpþqÞ

t¼ðj�1ÞðpþqÞþpþ1

�h�1s h�1t fa2jCovðKðx�Xs

hsÞ;Kðx�Xt

htÞÞj

þ b2jCovðKðy�Xs

hsÞ;Kðy�Xt

htÞÞjþab½jCovðKðx�Xs

hsÞ;Kðy�Xt

htÞÞj

þjCovðKðy�Xs

hsÞ;Kðx�Xt

htÞÞj�g

� B2ða2þb2þ2abÞ 1

nh3n½�CovðX 0ni;X

0njÞ�; ð2:15Þ

where X 0ni ¼PiðpþqÞ

s¼ði�1ÞðpþqÞþpþ1 Xs; X 0nj ¼PjðpþqÞ

t¼ðj�1ÞðpþqÞþpþ1 Xt: While, by sta-tionarity

jCovðX 0n1;X 0nðlþ1ÞÞj

¼ jXq

r¼1ðq� r þ 1ÞCovðX1;XlðpþqÞþrÞ þ

Xq�1

r¼1ðq� rÞCovðXrþ1;XlðpþqÞþ1Þj

¼ jXq

r¼1ðq� r þ 1ÞCovðX1;XlðpþqÞþrÞ þ

Xq�1

r¼1ðq� rÞCovðX1;XlðpþqÞ�rþ1Þj

�XlðpþqÞþq

r¼lðpþqÞ�ðq�2ÞjCovðX1;XrÞj ¼ q

XlðpþqÞþðq�1Þ

r¼lðpþqÞ�ðq�1ÞjCovðX1;Xrþ1Þj: ð2:16Þ

Hence, by (2.15), (2.16), (A4) and using stationarity again, we get

2X

1�i<j�k

jCovðy0ni; y0njÞj

� �2B2ða2 þ b2 þ 2abÞ 1

nh3n

Xk�1

l¼1ðk � lÞCovðX 0n1;X 0nðlþ1ÞÞ

� Ckqnh3

n

Xk�1

l¼1

XlðpþqÞþðq�1Þ

r¼lðpþqÞ�ðq�1ÞjCovðX1;Xrþ1Þj

� Ckqn� 1h3

n

X1

r¼p

jCovðX1;Xrþ1Þj ! 0;

which, together with (2.14) and (2.9), implies that VarðT 0nÞ ! 0:

Lemma 2.4 Under the assumptions of Theorem 1.1, we have VarðT 00n Þ ! 0:

Proof. Note that

VarðT 00n Þ ¼Xn

l¼kðpþqÞþ1VarðnnlÞ þ 2

X

kðpþqÞþ1�i<j�n

Covðnni; nnjÞ :¼ J1 þ J2:


Similarly to the proof in (2.12) and noticing n� kðp þ qÞ � p þ q � 2p, wehave

J1 � ½a2CðxÞ þ b2CðyÞ þ 2abC þ 2abhnCðx; yÞ� � n� kðp þ qÞn

� ½a2CðxÞ þ b2CðyÞ þ 2abC þ 2abhnCðx; yÞ� � 2pn! 0:

Similarly to the proof in (2.13), we have J2 � 8M0ða2 þ b2 þ 2abÞ � pn � phn ! 0:

Lemma 2.5 Under the assumptions of Theorem 1.1, we haveVarðSnÞ ! a2r2ðxÞ þ b2r2ðyÞ; further VarðTnÞ ! a2r2ðxÞ þ b2r2ðyÞ:

Proof. As in the proof of Lemma 2.3, we get

VarðTnÞ ¼Xk

m¼1VarðynmÞ þ 2

X

1�i<j�k

Covðyni; ynjÞ

� ½a2CðxÞ þ b2CðyÞ þ 2abC� � kpnþ 2abCðx; yÞ � kp

n� hn

þ 2M0ða2 þ b2 þ 2abÞ � kp2hn

nþ C � kp

n� 1h3

n

X1

r¼q

jCovðX1;Xrþ1Þj

<1: ð2:17ÞObviously,

VarðSnÞ ¼VarðTnÞ þ VarðT 0nÞ þ VarðT 00n Þþ 2CovðTn; T 0nÞ þ 2CovðTn; T 00n Þ þ 2CovðT 0n; T 00n Þ: ð2:18Þ

Since ETn ¼ ET 0n ¼ ET 00n ¼ 0, by Lemmas 2.3 and 2.4, (2.17) and using theCauchy-Schwarz inequality, we get

jCovðTn; T 0nÞj ! 0; jCovðTn; T 00n Þj ! 0; jCovðT 0n; T 00n Þj ! 0: ð2:19ÞNote that

VarðTnÞ þ VarðT 0nÞ þ VarðT 00n Þ

¼Xk

m¼1VarðynmÞ þ 2

X

1�i<j�k

Covðyni; ynjÞ

þXk

m¼1Varðy0nmÞ þ 2

X

1�i<j�k

Covðy0ni; y0njÞ þ

Xn

l¼kðpþqÞþ1VarðnnlÞ

þ 2X

kðpþqÞþ1�i<j�n

Covðnni; nnjÞ ¼Xn

i¼1VarðnniÞ þ 2An; ð2:20Þ

where

An¼P

1�i<j�k Covðyni; ynjÞþP

1�i<j�k Covðy0ni; y0njÞ þ

PkðpþqÞþ1�i<j�n Covðnni; nnjÞ

þ 12

Pkm¼1 I2ðmÞ þ

Pkm¼1

Pðm�1ÞðpþqÞþ1�i<j�ðm�1ÞðpþqÞþp

Covðnni; nnjÞ and by the

proof in Lemmas 2.3 and 2.4, we have


An ! 0: ð2:21ÞWhile

Xn

i¼1VarðnniÞ ¼

hn

n

Xn

i¼1h�2i ½a2VarðKðx� Xi

hiÞÞ þ b2VarðKðy � Xi

hiÞÞ

þ 2abCovðKðx� Xi

hiÞ;Kðy � Xi

hiÞÞ�

:¼ L1ðnÞ þ L2ðnÞ þ L3ðnÞ: ð2:22ÞBy Lemma 2.1 and using the Toeplitz lemma and (A3), we get

L1ðnÞ ! a2r2ðxÞ; L2ðnÞ ! b2r2ðyÞ; L3ðnÞ ! 0: ð2:23ÞTherefore, from (2.18)-(2.23) it follows that VarðSnÞ ! a2r2ðxÞ þ b2r2ðyÞ:Further, by Lemmas 2.3 and 2.4 and (2.18)-(2.19), we have

VarðTnÞ ! a2r2ðxÞ þ b2r2ðyÞ:

Lemma 2.6 Under the assumptions of Theorem 1.1, we have

jEeitPk

m¼1 ynm �Yk

m¼1Eeitynm j ! 0; 8t 2 <: ð2:24Þ

gnð�Þ ¼Xk

m¼1E½X 2

nmIðjXnmj � �Þ� ! 0; 8� > 0; ð2:25Þ

where Xnm ¼ ynm=sn; sn ¼Pk

m¼1 VarðynmÞ:

Proof. By Lemma 2.2 , using the monotonicity of hj and stationarity of Xj,we have

jEeitPk

m¼1 ynm �Yk

m¼1Eeitynm j

� jCovðeitPk�1

m¼1 ynm ; eitynk Þj þ jEeitPk�1

m¼1 ynm �Yk�1

m¼1Eeitynm j � � � �

� jCovðeitPk�1

m¼1 ynm ; eitynk Þj þ jCovðeitPk�2

m¼1 ynm ; eityn;k�1Þjþ � � � þ jCovðeityn2 ; eityn1Þj

� ½Bðjaj þ jbjÞ�2t2 � hn

n½X

j2I1

X

l2I2

h�2j h�2l jCovðXj;XlÞj

þX

j2ðI1þI2Þ

X

l2I3

h�2j h�2l jCovðXj;XlÞj

þ � � � þX

j2ðI1þI2þ��þIk�1Þ

X

l2Ik

h�2j h�2l jCovðXj;XlÞj�

� ½Btðjaj þ jbjÞ�2 � 1

nh3n½X

j2I1

X

l2I2

jCovðXj;XlÞj þX

j2ðI1þI2Þ

X

l2I3

jCovðXj;XlÞj


þ � � � þX

j2ðI1þI2þ��þIk�1Þ

X

l2Ik

jCovðXj;XlÞj�

� ½Btðjaj þ jbjÞ�2 � 1

nh3n½ðk � 1Þ

X

j2I1

X

l2I2

jCovðXj;XlÞj

þ ðk � 2ÞX

j2I1

X

l2I3

jCovðXj;XlÞj þ � � � þX

j2I1

X

l2Ik

jCovðXj;XlÞj� ð2:26Þ

Once again, by stationarity,

X

j2I1

X

l2Ik

jCovðXj;XlÞj ¼ pjCovðX1;Xðk�1ÞðpþqÞþ1Þj

þ ðp � 1ÞjCovðX1;Xðk�1ÞðpþqÞþ2Þj þ � � � þ jCovðX1;Xðk�1ÞðpþqÞþpÞj ð2:27Þ

Hence, from (2.26), (2.27) and (A4) it follows that

jEeitPk

m¼1 ynm �Yk

m¼1Eeitynm j � C

pkn� 1h3

n

X1

j¼p

jCovðX1;Xjþ1Þj ! 0 as n!1:

Thus, (2.24) is verified. Now, we prove (2.25)From (A2)(i) we know that jKðxÞj � C; x 2 <, and hence

jXnmj �1

snpffiffiffiffiffiffiffiffiffiffihn=n

p� h�1n ð2aC þ 2bCÞ � Cp

snffiffiffiffiffiffiffinhnp :

Therefore, by (A4)(ii),

gnð�Þ �Xk

m¼1

C2p2

s2n � nhnP ðjXnmj � �Þ �

C2

�2s2n� p2

nhn! 0:

Proof of Corollary 1.1. Note that Z�nðxÞ ¼ffiffiffiffiffiffiffinhnp

½fnðxÞ� EfnðxÞ�þffiffiffiffiffiffiffinhnp

½EfnðxÞ � f ðxÞ� and

ffiffiffiffiffiffiffinhn

p½EfnðxÞ � f ðxÞ� ¼

ffiffiffiffiffiffiffinhn

pð1n

Xn

j¼1h2

j Þð1

n

Xn

j¼1h2

j Þ�1½EfnðxÞ � f ðxÞ�

¼ffiffiffiffiffiffiffiffiffiffiffiffin�1hn

p Xn

j¼1h2

j � ð1

n

Xn

j¼1h2

j Þ�1½EfnðxÞ � f ðxÞ�:

Theorefore, Theorem 1.1, Lemma 1.1 andffiffiffiffiffiffiffiffiffiffiffiffin�1hn

p Pnj¼1 h2

j ! 0 imply theconclusion.

Acknowledgements. The authors thank the referee for carefully reading the manuscript and for

valuable suggestions which improved the presentation of this paper. This research was partially

supported by the National Natural Science Foundation of China (No. 10171079), No.R01-2000-

000-00010 and No. 2001-42-D0008 from Korea Research Foundation as well as Wonkwang

University Grant in 2003.


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asymptotic normality of recursive density estimates under some dependence assumptions

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