Joural of Multivariate Aalysis 70, 250285 (1999) Article ID jmva.1999.1822, available olie at httpwww.idealibrary.com o Coditioal Empirical Processes Defied by Nostatioary Absolutely Regular Sequeces Michel Harel ad Mada L. Puri* IUFM du Limousi, U.M.R.L. 55830 C.N.R.S., Toulouse, Frace, ad Idiaa Uiversity, U.S.A. E-mail hareluilim.fr Received May 1, 1997 K. I. Yoshihara (1990, Comput. Math. Appl. 19, No. 1, 149158) proved the weak ivariace of the coditioal earest eighbor regressio fuctio estimator called the coditioal empirical process based o.-mixig observatios. I this paper, we exted the result for ostatioary ad absolutely regular radom variables which have applicatios for Markov processes, for which the iitial measure is ot ecessary, the ivariat measure. 1999 Academic Press AMS 1991 subject classificatio 62G05. Key words ad phrases empirical distributio fuctio; coditioal empirical process; Skorohod topology; Gaussia process; absolute regularity. 1. INTRODUCTION Let [Z i =(X i, Y i ); i1] be a sequece of radom variables with cotiuous d.f.'s (distributio fuctios) H i (z), i1, z # R 2, ad we assume that H i (z) admits a strictly positive desity h i (z), i1, ad H i has the two margials F i ad G i where the respective desities are oted by f i ad g i. Suppose that there exists a radom vector (X, Y) ir 2 with fiite expectatio E Y ad which admits H as distributio fuctio, the regressio fuctio m(x)=e(y X=x), x # R of Y o X is a.s. i x uiquely defied i view of the equatio m(x)=e(y X). (1.1) Suppose also that H i coverges to the distributio fuctio H which admits a desity h ad H has the two margials F ad G where the respective desities are oted by f ad g. Cosider the earest eighborhood estimate m~ (x 0 )=(a ) 0047-259X99 30.00 Copyright 1999 by Academic Press All rights of reproductio i ay form reserved. (x 0 )&F (X i ) Y i K \F (1.2) a * Research supported by the Office of Naval Research Cotract N00014-91-J-1020. 250
CONDITIONAL EMPIRICAL PROCESSES 251 of m(x 0 ) where K is a appropriate kerel fuctio, a is a sequece of badwidths such that a 0 as, (1.3) where F is the empirical distributio fuctio of X 1,..., X. Yoshihara (1988) uder some coditios showed whe the r.v.s. are. mixig ad statioary that for some _ 2 0 (a ) 12 [m~ (x 0 )&m(x 0 )] coverges to the ormal law with mea 0 ad variace _ 2 0 x 0 # R such that f(x 0 )>0. Let for almost all m( y x)=p(y y X=x), (x, y)#r 2 (1.4) be the coditioal distributio fuctio ad as a estimator of m( } x) we cosider the coditioal empirical process m ( y x 0 )=(a ) (x 0 )&F (X i ) I [Yi y]k \F, y # R, (1.5) a where F is the empirical distributio fuctio of X 1, X 2,..., X. The mai result of this paper is to show that uder some coditios (a ) 12 [m (G (}) x 0 )&m(g (}) x 0 )] coverges weakly i Skorohod topology to B 0 where B 0 is a certai Gaussia process (depedig o x 0 ). Stute (1986) showed this result whe the r.v.'s (radom variables) are idepedet ad idetically distributed, the Yoshihara (1990) geeralized it for.-mixig ad idetically distributed r.v.'s. The geeralizatio for absolutely regular ad ostatioary r.v.'s give applicatios for the importat case of Markov processes, for which the iitial measure is ot ecessary, the ivariat measure. Examples of models are give i Sectio 2. For the particular case where Y i #X i1, the coditioal empirical process m ( ) has the expressio m ( y x 0 )=(a ) (x 0 )&F (X i ) I [Xi1 y]k \F a. (1.6) Recall that the process satisfies the absolute regularity coditio if max j1 E[ sup P(A _(Z i,1ij))&p(a) ] A # _(Z i, i jm) =;(m) a 0 as m. (1.7)
252 HAREL AND PURI Here _(Z i,1i j) ad _(Z i, i jm) are the _-fields geerated by (Z 1,..., Z j ) ad (Z jm, Z jm1,...), respectively. Also recall that [Z i ] satisfies the strog mixig coditio if max[sup [ P(A & B)&P(A) P(B) ; A # _(Z i,1ij), ] j1 B # _(Z i, i jm)]=(m) a 0 as m ad [Z i ] satisfies the.-mixig coditio if max[sup [ P(A B)&P(A) ; B # _(Z i,1ij), A # _(Z i, i jm)]] j1 =.(m) a 0 as m. Sice (m);(m).(m), it follows that if [Z i ] is absolutely regular, the it is also strog mixig ad if [Z i ] is.-mixig, it is also absolutely regular. 2. CONVERGENCE OF THE CONDITIONAL EMPIRICAL PROCESS Deote F ad G by For ay i # N, put Let us deote F = F i, G = V i (u, v)=h i (F (u), G (v)). H (x, y)= I [Xi x, Y i y] the empirical distributio fuctio of (X 1, Y 1 ), (X 2, Y 2 ),..., (X, Y ) ad V (u, v)= G i. I [F (X i )u, G (Y i )v] the empirical distributio of (F (X 1 ), G (Y 1 )),..., (F (X ), G (Y )). We have E(V (u, v))=v (u, v)= V i (u, v).
CONDITIONAL EMPIRICAL PROCESSES 253 Deote also M (u)= I [F (X i )<u] ad M (u)=e(m (u)), the we may write m ( y x 0 )=a I (x 0 )&F (x) (&, y](z) K \F a H (dx, dz) =a I [0, G (v) K (F (x 0 ))&M (u) ( y)] \M a V (du, dv) =m~ (G ( y) F (x 0 )). (2.1) Let us deote m~ (v u 0 )=m(g (v) F (u 0 )). (2.2) I the followig, we suppose that the sequece [Z i =(X i, Y i ), i1] satisfies ad further we suppose also that sup E( Y i 5$ )< for some $>0 (2.3) i1 sup m~ (v u)&m~ (w u) =o((log $ ) ) as $ 0 (2.4) v&w $ uiformly i a eighborhood of u 0. For each v, m~ (v } ) (defied i (2.2)) is twice cotiuously differetiable i a eighborhood V of u 0 such that sup sup m~ "(v u) <. (2.5) u # V v K is a twice cotiuously differetiable kerel fuctio o R such that K(u)=0 for u 1, K(u) du=1 ad uk(u) du=0 (2.6) ad a 4, a5 0. (2.7) As usual, let D(0, 1) be the space of all right-cotiuous fuctios o [0, 1] with left had limits. Cosider a sequece [Z i ] i1 of real radom vectors which is a Markov process with trasitio probability P(z; A) where z=(x, y)#r 2 ad A # B, B is the Borel _-field of R 2.
254 HAREL AND PURI Recall that the Markov process is geometrically ergodic if it is ergodic ad if there exists 0<\<1 such that &P (z; })&(})&=O(\ ) for all a.s. z # R 2, (2.8) where is the ivariat measure ad P the -step trasitio probability. We say that the process [Z i ] i1 has & for the iitial measure if the law of probability of Z 1 is defied by & ad for ay i>1, the law of probability P i of Z i is defied by &P i. For ay probability measure & ad ay trasitio probability Q we deote by Q& the probability measure defied o R 4 by Q&(A_B)= B Q(z; A) &(dz) for ay A_B # B_B. The Markov process is called strogly aperiodic if for ay z # R, the trasitio probability P(z; } ) is equivalet to the Lebesgue measure. The Markov process is called Harris recurret if there exists a _-fiite measure & o R 2 with &(R 2 )>0 such that &(A)>0 implies P(z; Z i # Ai.o.)=1 for all z # R 2. Theorem 2.1. Let [Z i ] i1 be a Markov process which is strogly aperiodic, Harris recurret ad geometrically ergodic. Suppose that the ivariat measure has a cotiuous desity h ad the trasitio probability P has a cotiuous desity p(}; }) ad furthermore suppose that the coditios (2.3)(2.5) are satisfied. The for ay iitial measure, ad for almost all 0<u 0 <1, (a ) 12 [m (G (v) F (u 0 ))&m~ (v u 0 )] coverges i law to a Gaussia process B 0 (u 0 ) as with respect to the Skorohod topology o D(0, 1) where B 0 is cetered, has cotiuous sample paths vaishig at the lower boudary of [0, 1] ad covariace [m~ (v 1 7 v 2 u 0 )&m~ (v 1 u 0 ) m(v 2 u 0 )] K 2 (u) du. (2.9) Example 2.1. We cosider a ARMA process X i =ax i b= i = i, i # N*, (2.10) where X 0 admits a strictly positive desity, [= i ] is a sequece of i.i.d. real valued radom variables with strictly desity such that E(= i )=0, ad a ad b are real umbers such that a <1. The the coditios of Theorem 2.1 are satisfied for the process defied i (2.10), because we have the particular case where Y i #X i1. The law of the process o which observatios are take is defied by the iitial measure (i.e. the measure which defies
CONDITIONAL EMPIRICAL PROCESSES 255 the lax of X 0 ) ad the trasitio probability. We ca estimate the coditioal distributio fuctio by the coditioal distributio process m ( ) for ay iitial measure of X 0 which admits a strictly positive desity. Theorem 2.2. Let [Z i ] i1 be a Markov process which is aperiodic ad Doebli recurret. Suppose that the ivariat measure has a cotiuous desity fuctio h ad the trasitio probability P has a coditio desity p( ;) ad furthermore the coditios (2.3)(2.5) are satisfied. The for ay iitial measure, the coditios of Theorem 2.1 also hold for the coditioal distributio process m. Example 2.2. We cosider the process X i = f(x i )= i, i # N*, (2.11) where X 0 admits a strictly positive desity ad [= i ] i1 is a sequece of i.i.d. real radom variables with strictly positive desity such that E(= i )=0 ad f is a bouded fuctio. That is a particular case of Markov process satisfyig the coditios of Theorem 2.2 where Y i #X i1. Now more geerally, cosider the model Y i =(X i )e i, i1, where X i deotes a radom variable of observed values, is a cotiuous kow fuctio such that (2.3) is satisfied, e i is a white oise of idetically absolutely regular radom variables with a geometrical rate ad strictly positive desity ad Y i is a predictor variable. For ay iitial measure of [(X i, Y i )] i1, the coclusio of Theorem 2.1 holds for the coditioal empirical process defied i (2.1) whe the process [X i ] is a process of the Example 2.1 or the Example 2.2; defied i (2.10) or (2.11). I fact, the Theorems 2.1 ad 2.2 will be deduced from the followig theorem proved uder a broader framework. We suppose that the sequece [Z i =(X i, Y i ), i1] is absolutely regular with rates ;(m)=o(\ m ), 0<\<1. (2.12) We say that a sequece of d.f.'s [H l *] o R 4 admits F l *, G l * ad H as margials o R 2 ad F ad G o R if we have lim H l *(x, y, u, v)=f l *(x, u) y v lim x u H l *(x, y, u, v)=g l *( y, v)
256 HAREL AND PURI lim H l *(x, y, u, v)=h(x, y) u v lim x y H l *(x, y, u, v)=h(u, v) ad the margials of H are F ad G. PutH i, j, the distributio fuctio of (Z i, Z j ). Theorem 2.3. Suppose that the sequece [Z i ] is absolutely regular with rates (2.12) ad satisfyig (2.3). Furthermore, assume that for ay l>1, there exists a cotiuous d.f. H l * o R 4 admittig a desity h l * with margials F l *, G l * ad H o R 2 ad F ad G o R such that the desity of H is bouded i a eighborhood of F (u 0 )_R ad &H i, j &H* j&i &=O(\ i 0 ), 1i< j, 1, 0<\ 0<1 (2.13) for which there exists a sequece [Z i=(x i, Y i)] of statioary radom variables such that [Z i] is absolutely regular with rates (2.12) ad (Z i, Z j) has H* j&i as d.f. (i< j1). Suppose also that the coditios (2.3)(2.5) are satisfied. The for almost all 0<u 0 <1, the coclusios of Theorem 2.1 hold for the coditioal distributio process m. Remark 2.1. I Yoshihara (1988), the coditio (2.7) was replaced by the weaker coditio a 3 ad a 5 0, but we have to stregthe his coditio to obtai the geeralizatio to the absolutely regular ad ostatioary case (see the Proof of Lemma 4.3). We deote _ 2 0 =Var(Y X=x 0) K 2 (u) du, (2.14) where (X, Y) is a radom vector which admits H as a distributio fuctio. The Proof of Theorem 2.1 eeds the followig propositios. Propositio 2.1. Uder the coditios of Theorem 2.1, the (a ) 12 [m~ (x 0 )&m(x 0 )] coverges to the ormal law with mea 0 ad variace _ 2 0 for almost all x 0 # R such that f(x 0 )>0. Propositio 2.2. 0<u 0 <1, Uder the coditios of Theorem 2.1, for almost all (a ) 12 [m~ (v u 0 )&m~ (v u 0 )] coverges i law to the Gaussia process B 0 (u 0 ) as with respect to the Skorohod topology where m~ is defied i (2.1).
CONDITIONAL EMPIRICAL PROCESSES 257 3. PROOFS OF THEOREMS 2.1, 2.2, AND 2.3 3.1. Proof of Theorem 2.1 From Theorem 2.3, we have oly to prove (2.12) ad (2.13). First, we prove (2.12). From Davydov (1973), ad the coditio of strog aperiodicity, we have ;(m)=sup P (dz) &P m (z; })&P m (})& sup P (dz) &P m (z; })&(})&&P m (})&(})&. As the process is geometrically ergodic, we ca fid 0<\<1 such that &P m (z; })&(})&=O(\ m ) for all a.s. z # R 2. (3.1) From Theorem 2.1 of Nummeli ad Tuomie (1982), we deduce which is (2.12). Now, we prove (2.13). We have from (3.1) that is, &P m P &P m & ;(m)=o(\ m ) (3.2) =2 sup P A_B # B_B} m (z; A) P B (dz)& P m (z; A) (dz) } B 2 sup P m (z; A) } P (dz)&(dz) } A_B # B_B B 2 &P &&=O(\ ), Thus the coclusios of Theorem 2.1 holds. &P m P &P m &=O(\ ). (3.3) 3.2. Proof of Theorem 2.2 From Theorem (4.1) of Davydov (1973) the process [X i ] is geometrically. mixig which implies geometrically absolute regularity. The proof is ow similar to Theorem 2.1.
258 HAREL AND PURI 3.3. Proof of Theorem 2.3 By defiitio, we have m (G (v) F (u 0 ))&m~ (v u 0 ) =m~ (G b G (v) F b F (u 0 ))&m~ (v u 0 ) =m~ (G b G (v) F b F (u 0 ))&m~ (G b G (v) u 0 ) m~ (G b G (v) u 0 )&m~ (v u 0 ). By Propositio 2.2, we kow that (a ) 12 [m~ (v u 0 )&m~ (v u 0 )] coverges i law to B 0 (u 0 ). We deduce from Theorem 5.5 of Billigsley (1968) that coverges also i law to B 0 (u 0 ). We ca write (a ) 12 [m~ (G b G (v) u 0 )&m~ (v u 0 )] m~ (G b G (v) F b F (u 0 ))&m~ (G b G (v) u 0 ) =a I [O, G (v)](w) b G _ {K \ M (F b F (u 0 ))&M (F b F (u)) a (u 0 )&M (u) &K \M a = G (du, dw)=i. From (2.13) ad usig the fact that K is twice cotiuously differetiable, we deduce that I =o p (1). 4. PROOF OF PROPOSITION 2.1 From ow o, the letter C, with or without subscript, will deote some positive quatity. Writig K(F (x 0 )&F (X i )a )as K[(F (x 0 )&F (X i )F (x 0 )&F (X i )&F (x 0 )F (X i ))a ]
CONDITIONAL EMPIRICAL PROCESSES 259 ad usig Taylor's expasio, we get m~ (x 0 )=(a ) Y i K \F (x 0 )&F (X i ) a a &2 Y i [F (x 0 )&F (X i )&F (x 0 )F (X i )] _K$((F (x 0 )&F (X i ))a ) a &3 Y i [F (x 0 )&F (X i )&F (x 0 )F (X i )]K"(2 i ) =I 1 I 2 I 3, (4.1) where 2 i is some radom umber betwee a [F (x 0 )&F (X i )] ad a [F (x 0 )&F (X i )]. We eed the followig lemmas. Lemma 4.1. Uder the coditios of Theorem 2.3, (a ) 12 I 3 w p 0 as. (4.2) Proof. Sice K vaishes outside (, 1), the above expasio of m (x 0 ) holds true with itegratio restricted to those x for which F (x 0 )&F (x) <a.letb =(a 3 ) 13. The, by Lemmas 6.7 ad 6.8 P[ sup (a ) 12 F (x 0 )&F (x) x F (x 0 )&F (x) C 1 a &F (x 0 )F (x) b ] =P[ sup x F (x 0 )&F (x) C 1 a M *(F (x 0 ))&M *(F (x)) b a 12 ] Cb &4 =C(a 3 )&23, (4.3) where C is some costat >0. O the other had, as [Y i ] is strog mixig with rates (2.12) ad coditio (2.3) is satisfied, we obtai E { \ Y i &E(Y i ) = 2 C 2, (4.4) where C 2 is some costat >0. From the Markov iequality, we deduce that for ay =>0 P _} Y i &E \ Y i} >= & =2 C 2 (4.5)
260 HAREL AND PURI which implies that Y i&e( Y i) w p 0 ad we have E( Y i) E(Y)<. Thus Y i w p E(Y)<. (4.6) Thus, for a arbitrary umber =>0, ad ay positive umber ', we ca fid sufficietly large such that where P(B ('))1&=, B (')= {, E(Y)&' Y i E(Y)' =. From (4.3) ad the fact that K" is bouded, upo observig that a 3 we ca fid a iteger sufficietly large such that o the set B (') for ay $>0 As $ is arbitrary, (4.2) follows. (a ) 12 I 3 <$. (4.7) Deote m i (x)=e(y i X i =x) ad m (x)= m i (x), x # R. Lemma 4.2. Uder the coditios of Theorem 2.3, (a ) 12 I 2 is asymptotically equivalet to &(a 2 m (x 0 ) 2 ) {\ (x 0 )&F (X i ) K \F a & \ E \ F (x 0 )&F (X i ) \K a =. (4.8) Proof. Defie [Z, 1] by Z = a &32 (Y i &m i (X i ))(W (x 0 ) &W (X i )) K$ (x 0 )&F (X i ) \F a,
CONDITIONAL EMPIRICAL PROCESSES 261 where W (x)= 2 (I [Xi x]&f i (x)), x # R. (4.9) Proceedig as i Lemma 4.2 of Yoshihara (1988), we prove that Now cosider the fuctio Z w p 0 as. (x 0 )&F (x) k (x, y)=m (x) K$ \F a (I [ yx 0 ]&I [ yx] ) ad the radom variable T = k (x, y) W (dx) W (dy). The, usig the method of the Proof of Lemma 2.2 i Harel ad Puri (1989) ad usig (2.12), we obtai which implies E(T 2 )=O(1) as a &32 k (x, y) W (dy)[f (dx)&f (dx)] w p 0 sice a 3. Thus, we obtai that (a ) 12 I 2 is asymptotically equivalet to a &32 Now we prove that m (x 0 )&F (x) (x)(w (x 0 )&W (x)) K$ \F a F (dx). for a.s. x 0 # R. I 4 =a &32 m (x)&m (x 0 ) W (x 0 )&W (x) _ } K$ \ F (x 0 )&F (x) a } F (dx) w p 0 (4.10)
262 HAREL AND PURI From F (F (x 0 ))=x 0 for a.s. x 0 # R ad ay # N*, we obtai lim 1 m (F for a.s. s # R. Put e (x 0 )= \a 1 m (F 0 (s)&ua )&m (F (s)) K$(u) du=0 (u))&m (x 0 ) } K$ \ F (x 0 )&u a } The, by Lemma 6.8, we have that for ay =>0, P(I 4 >=)P \a&32 sup W (F (x 0 ))&W (u) F (x 0 )&u a 6 du. _ 1 m (F (u))&m (x 0 ) } K$ 0 \ F (x 0 )&u a } du>= P( sup W (F(x 0 ))&W (u) e (x 0 ) a 12 ) F (x 0 )&u a Ce &4 (x 0 )=o(1). Hece (4.10) is proved ad it follows that (a ) 12 I 2 is asymptotically equivalet to a &32 m(x 0 ) [W (x 0 )&W (x)] K$ \F (x 0 )&F (x) a =&a &32 =&a 2 which leads to (4.8). m(x 0 ) [W (x)] K$ \F (x 0 )&F (x) a m(x 0 ) K \ F (x 0 )&F (x) a W (dx) Lemma 4.3. Uder the coditios of Theorem 2.3 I 4 =(a ) 2 &(a ) 2 (Y i &m i (x 0 )) K \F (x 0 )&F (X i ) a F (dx) F (dx) (x 0 )&F (X i ) E {(Y i&m i (x 0 )) K \F (4.11) a = has asymptotically (as ) the ormal distributio with mea 0 ad variace _ 2 0 (deoted as N(0, _2 0 )).
CONDITIONAL EMPIRICAL PROCESSES 263 Proof. ad Let B i =a 2 (Y i&m(x 0 )) K \F(x 0)&F(X i) a &a 2 a = E {(Y i&m(x 0 )) K \F(x 0)&F(X i) B* i =a 2 (Y i &m i (x 0 )) K \F (x 0 )&F (X i ) a &a 2 a = E {(Y i&m i (x 0 )) K \F (x 0 )&F (X i ) B i =a 2 (Y i &m(x 0 )) K \F(x 0)&F(X i ) a &a 2 The, we have a = E {(Y i&m(x 0 )) K \F(x 0)&F(X i ) I 4 = 2 From (2.13), it follows that B* i = 2 which alog with (2.3) implies that or We deduce that E } 2 B i 2,,...,, sup F(x)&F (x) =O( ) x # R E B* i &B i =O(a &32 ) 2 (B i &B* i ) }=O(a&32 2 ). (B i &B* i ) w p 0.,,...,,,,...,. (B* i &B i ).
264 HAREL AND PURI Now, we have to show that 2 Let us ow deote B i coverges i law to N(0, _ 2 0 )..(x, y)=(y&m(x 0 )) K \F(x 0)&F(x) a For ay M>0, deote also by & (z&m(x 0)&F(u) 0)) K \F(x H(du, dz). a B M i =a2 (Y i I [ Yi M]&m(x 0 )) K \F(x 0)&F(X i ) a B &a 2 E {(Y ii [ Yi M]&m(x 0 )) K \F(x 0)&F(X i ) a = M i =B i&b M i. M (x, y)=(i [ y M] y&m(x 0 )) K \F(x 0)&F(x) a & (I 0)&F(u) [ z M]z&m(x 0 )) K \F(x H(du, dz) a. M (x, y)=.(x, y)&. M (x, y) (i)=2a. l(x, y). p (u, z) H*(dx, i dy, du, dz), M, l, p D 0l, p1, i1 M, l, p D (0)=. l(x, y). p (x, y) H(dx, dy), 0l, p1, where. 0 (x, y)=. M (x, y),. 1 (x, y)=. M (x, y) D (i)= 0l, p1 M, l, p D (i), i0.
CONDITIONAL EMPIRICAL PROCESSES 265 We the have } E(B i B i )& (&i) D (i) } 1i, j = } [E(B M )E(B M i i )][E(BM)E(M M )] j j 1i, j & 0l, p1 From coditio (2.3), we have M, l, p (&i) D (i) }. sup 1 M max E(B i )2 =O(M &2$ ) 1i ad We ca write } 1i< j (. M (x, y)) 2 H(dx, dy)=o(m &2$ ). (4.12) E(B M i BM j )& (&i) M,0,0 (i) } k E(B } M i BM)& j (&i) D M,0,0 (i) } 1 j&ik E(B } M i BM)& j (&i) D M,0,0 (i) } k< j&i i=k1 =I (1) (1) J. From coditio (2.13) ad the fact that K$ is bouded, if we take M=a, ad from coditio (2.12) If we take k=a,the I (1) I (1) =ko(a&3 ) J (1) =O(\k 1 ), where \ 1=\ $(2$). (1) J =O(a&4 We deduce similarly that } )O(a \ a 1 )=O(a &4 ). (4.13) E(B M i )2 &D M,0,0 (0) }=O( a &4 ). (4.14)
266 HAREL AND PURI We ow write } M E(B i BM j )& (&i) D M,1,0 (i) } 1i< j } k M E(B i BM)& j (&i) D M,1,0 (i) } 1 j&ik } M E(B i B M j )& (&i) D M,1,0 (i) } k< j&i i=k1 =I (2) J (2). (4.15) From (4.12), we have ad from coditio (2.12) I (2) =O(ka$&2 ), J (2) =O(\k 1 ). If we take k=a ad otig that M=a, we have ad I (2) J (2) =O(a 2$&4 Similarly, we ca show that } E(B M B M i 1i, j } )O(a \ a 1 )=O(a $&3 ). (4.16) M E(B i B M i )&D M,1,0 (0) }=O(a$&3 ), (4.17) j )& } M E(B B M i j )& 1i, j We deduce from (4.13), (4.14), (4.16)(4.19) that Hece } 1i, j lim } () 1i, j E(B i B j )& (&i) D M,0,1 (i) }=O(a$&3 ), (4.18) (&i) D M,1,1 (i) }=O(a2$&2 ). (4.19) E(B i B j )&() (&i) D (i) }=O(a$&3 ). (&i) D (i) }=0 (4.20)
CONDITIONAL EMPIRICAL PROCESSES 267 which implies that lim } E \ 2 Now, we prove that 2 B i lim where _ 2 0 is defied by (2.14). We have E(B i) 2 =a y&m(x 0) 2 K 2 &a =a 2 &E B i } =0. \2 2 E B =_ i 2, (4.21) \2 0 \ F(x 0)&F(x) H(dx, dy) a { ( y&m(x 0)) K \F(x 0)&F(x) a a (x) K 2\ F(x 0)&F(x) &a { 1 = 1 0.(x) K \F(x 0)&F(x) a F(dx) (F (F(x 0 )&a u) K 2 (u) du) 2 F(dx) = 2 H(dx, dy) = &a {.(F (F(x 0 )&a u) K(u) du) = 2, (4.22) where (x)=e[ X 2 &m(x 0 ) 2 X 1 =x] ad.(x)=e[x 2 &m(x 0 ) X 1 =x]. From (4.22), we easily deduce that Now, we ca write lim E(B i) 2 =_ 2 0. (4.23) 2 E B = i \2 = 2 E(B i) 2 1i{ j E(B i) 2 2 1i< j j&i>k 1i< j j&ik E(B ib j) E(B ib j) E(B ib j). (4.24)
268 HAREL AND PURI The, we have E(B ib j)=a.(x, y) g(u, z) H* j&i(dx, dy, du, dz) Ca {.* i, j(x, u) K \F(x 0)&F(x) a 0)&F(u) _K \F(x a F* j&i(dx, du) = =Ca {.* i, j(f (F(x 0 )&a v), F (F(x 0 )&a w)) _ f * j&i(f (F(x 0 )&a v), F (F(x 0 ))&a w) K(v) f b F (F(x 0 )&a v) f b F (F(x 0 )&a w) _K(w) dv dw = Ca. (4.25) This follows from (2.3) ad the fact that f is cotiuous i a eighborhood of x 0 with f(x 0 )>0, where.* i, j (x, u)=e[(y j&m(x 0 ))(Y i&m(x 0 )) X i=x, X j=u] ad f * j&i is the desity of F* j&i. For ay $0, E B i 2$ Ca &(2$)2 y&m(x 0) } 2$ K \ F(x 2$ 0)&F(x) a } _H(dx, dy) =Ca &(2$)2 h*(x) } K \ F(x 2$ 0)&F(x) F(dx) a } Ca &$2 1 h*(f (F(x 0 )&a u)) K(u) 2$ du Ca &$2, (4.26) where h*(x)=e[ Y 1&m(x 0 ) 2$ X 1=x].
CONDITIONAL EMPIRICAL PROCESSES 269 From (4.25) ad (4.26), it follows that E \ 1i{ j B ib j 2Cka 2(Ca &$2 ) 2(2$) [;(i)] $(2$). (4.27) i=k1 If we take k=[a 2 ], we get (4.21) from (2.12) ad (4.23). By usig (4.26) ad Lemma 6.9 which geeralizes Lemma 3.4 of Yoshihara (1988) from the. mixig case to the strog mixig case, we obtai the followig iequality which geeralizes Lemma 5.1 of Yoshihara (1988) to the strog mixig case. E } m 2$ B i} Cm (2$)2 sup E B i 2$, m. (4.28) 1im Now we ca proceed as i Yoshihara (1988) (Proof of Theorem 2). Let p=[ 23 ], q=[ 13 ]adk=[( pq)]. Put The, we have ( j)(pq)p ' j = i=( j)(p1)1 ( j)(pq)p % j = % k1 = 2 i=( j)(pq)p1 i=k( pq)1 From (4.28), we deduce that B i. k B i = 2 k1 2 j=1 B i, j=1,..., k. j=1 B i, j=1,..., k. k1 ' j 2 % j w p 0. To prove Lemma 4.3, it remais ow to show that 2 k ' j=1 j coverges i law to N(0, _ 2 0 ) radom variable. From Lemma 6.1 we obtai E {exp \ it2 k j=1 ' j= & `k j=1 j=1 % j. [E[exp(it 2 ' j )]]Ck;(q),
270 HAREL AND PURI where C is some costat >0. Hece it is eough to prove that k ` j=1 Usig (4.28), we obtai [E[exp(it 2 ' j )]] coverges to e &(t2 2) _ 2 0. (4.29) E[exp(it 2 ' j )]=1& t2 2$ 2 E(' t j) 2 O \ 2$ E ' (2$)2 j From the fact that a 4 ad lim we get (4.29). The proof follows. =1& t2 2 E(' j) 2 o( t 2$ 6(2$) a &$2 ). ke(' j ) 2 =_ 2 0 (from (4.21)) Lemma 4.4. Uder the coditios of Theorem 2.3. (a ) _{ 12 yk \ F (x 0 )&F (x) a F i, i1(dx, dy) =&m(x 0) & 0 as 0. (4.30) Proof. We have D =(a ) _ 12 & (a ) 12 & (a ) 12 =H (1) yk \ F (x 0 )&F (x) a H i(dx, dy) yk \ F(x 0)&F(x) a _ H i(dx, dy) & yk \ F(x 0)&F(x) a yk \ F(x 0)&F(x) a _ yk \ F(x 0)&F(x) H i(dx, dy) H(dx, dy) & a H(dx, dy)&m(x 0) & (2) (3) H H. (4.31)
CONDITIONAL EMPIRICAL PROCESSES 271 From (2.3) ad (2.13), it follows that Deote ad We the have H (2) H (1) =O(a&32 2 ) (4.32) l(x, y)=yk 0)&F(x) \F(x a, 0)&F(x) l (M) (x, y)=i [ y M] yk \F(x a, l (M) (x, y)=l(x, y)&l (M) (x, y) for ay M>0. 12_ =(a ) l(m) (x, y)(h i (dx, dy)&h(dx, dy)) \ l (M) (x, y) H i (dx, dy)& l (M) (x, y) H(dx, dy) &. From (2.3) ad (2.13), if we take M= 14, we get H (2) =O( 4 a 2 )O( &(18)&($4) a 2 ). (4.33) From Stute (1984, Corollary, p. 919), it follows that H (3) 0 as. (4.34) Equatios (4.32)(4.34) yield (4.30). The Proof of the Propositio 2.1 ow follows from Lemmas 4.1 to 4.4. 5. PROOF OF PROPOSITION 2.2 Put ; (v)=; (v u 0 )=(a ) 12 [m~ (v u 0 )&m~ (v u 0 )]. (5.1) We shall prove Propositio 2.2 by showig the followig facts (i) the fiite-dimesioal distributios of ; coverge to those of B 0
272 HAREL AND PURI (ii) [; 1] is uiformly C-tight, i.e., for each =>0 ad every \>0, there exists some $>0 ad 0 such that for all 0 P( sup ; (v 1 )&; (v 2 ) \)=. (5.2) v 1 &v 2 $ The assertio (i) easily follows from Propositio 2.1 ad the Crame r-wold device. It remais to show (ii). Put (u 0 )&M (u) K (u)=k \M a V(u, v)=h(f (u), G (v)) ad Write m~ *(w u 0 )=a I [0, w](v) K (u) G(du, dv). ; (v u 0 )=(a ) 12 [m~ (v u 0 )&m~ *(v u 0 )] (a ) 12 [m~ *(v u 0 )&m~ (v u 0 )] =; 1 (v u 0 ); 2 (v u 0 ). (5.3) We show that both [; 1 ] ad [; 2 ] are uiformly C-tight. First, we ote that upo itegratig by parts, we have (u m~ (v u 0 )&m~ *(v u 0 )=a 0 ) [V (1, v)&v(1, v)] K \M a &a [V (u, v)&v(u, v)] K (du). (5.4) Sice by Lemma 6.6 i the Appedix M (u 0 ) u 0 (0<u 0 <1) with probability oe, a 0 ad K has fiite support, the first summad is zero with probability oe for all 0, say, ot depedig o v. Similarly, for 1 [V (u, v)&v(u, v)] K (du) = [V (u, v)&v (u, v)&v (u 0, v)v (u 0, v)] K (du) [V (u, v)&v(u, v)&v (u 0, v)v(u 0, v)] K (du). (5.5)
CONDITIONAL EMPIRICAL PROCESSES 273 Sice by assumptio K=0 outside (, 1), the last itegral remais uchaged whe restrictig the domai of itegratio to those u's for which M (u)&m (u 0 ) a. Let 0<=<1 be give arbitrarily. The by Lemma 6.7, we have, up to a evet of probability less tha or equal to =, that u 0 &u c 1 a, wheever M (u 0 )&M (u) a. Hece from (2.13) ad the boudaries of the desity of H, eglectig a evet of probability =, we obtai for all large that sup ; 1 (v 1 u 0 )&; 1 (v 2 u 0 ) a 2 &K& w *($)o p (1), (5.6) v 1 &v 2 $ where &K& deotes the total variatio of K ad w * is defied i Lemma 6.5. Cosequetly, the uiform C-tightess of [; 1, 1] follows from (6.11)(b) of Lemma 6.5 ad (5.6). As to ; 2, usig Taylor's expasio, we write m~ *(w u 0 )=a a &2 I 0&u [0, w](v) K \u V(du, dv) a I [0, w](v)[m (u 0 )&M (u)&u 0 u] 0&u _K$ \u V(du, dv) a 1 2 a&3 I [0, w](v)[m (u 0 )&M (u)&u 0 u] _K"(2) V(du, dv) =a m~ (w u 0 )I 2 (w, )I 3 (w, ), (5.7) where 2 is some costat betwee a [M (u 0 )&M (u)] ad a [u 0 &u]. By Lemma 4.1, we get (a ) 12 I 3 (w, ) w p 0. (5.8)
274 HAREL AND PURI Hece, to prove the assertio (ii) it remais to show that [(a ) 12 I 2 (, ) 1] is uiformly C-tight. We have (a ) 12 I 2 (w, ) =a &32 =a &32 a &32 m~ (w u)[u 0&u (u 0 )&U (u)] K$ \u a du [m~ (w u)&m~ (w u 0)][U (u 0 )&U (u)] K$ 0&u \u a du m~ (w u 0 ) [U 0&u (u 0 )&U (u)] K$ \u du, (5.9) a where U (u)= 12 M (u). It follows from (4.10) that the first summad coverges to zero i probability uiformly i y wheever m~ (} u) is equicotiuous i a eighborhood of u 0. Fially for large a &32 m~ (w u 0 ) [U 0&u (u 0 )&U (u)] K$ \u a du =&a 2 m~ (w u 0 ) K$ \ u 0&u a du (u) =&m~ (w u 0 )(a ) 2 _ { K \u 0&F (x i ) a From Propositio 2.1 we deduce that (a ) 2 { &E \ K \ u 0&F (X i ) a = 0&F (X i ) K \u a &E \ K \ u 0&F (X i ) a =. (5.10) coverges i law to the ormal distributio with mea 0 ad covariace u 0 K 2 (u) du ad so is stochastically bouded. Now C-tightess of ; 2 follows from this fact ad uiform cotiuity of m~ (} u 0 ). Propositio 2.2 is proved.
CONDITIONAL EMPIRICAL PROCESSES 275 6. APPENDIX Let [X i,1i, 1] be a ostatioary sequece of r.v.'s. Let p1 ad 1i 1 <i 2 <}}}<i p be arbitrary itegers. For ay j(1 jp), i1,..., ip Pj, is the probability measure defied by P i 1,..., i p j, (A ( j) _B ( p& j) ) =P[(X i1,..., X ij )#A ( j) ]_P[(X ij1,..., X ip )#B ( p& j) ] ad P i 1,..., i p o, is the probability measure defied by P i 1,..., i p o, (A ( p) )=P[X i1,..., x ip # A ( p) ] for ay A ( j) # _(X i1,..., X ij )(1jp) ad ay B ( p& j) # _(X ij1,..., X ip ) (1 jp). We state the followig two lemmas, the proof of the first oe follows essetially by proceedig as i Lemma 1 of Yoshihara (1976) ad the secod lemma is from Doukha ad Portal (1987, Propositio 2.8). Lemma 6.1. For every p1 ad (i 1,..., i p ) such that i 1 <i 2 <}}}<i p ad ay j (0 jp), let h(x 1,..., x p ) be a Borel fuctio such that R p h(x 1,..., x p ) 1$ dp (i 1,..., i p ) j, M for some $>0. The } R p h(x 1,..., x p ) dp (i 1,..., i p ) o, & h(x 1,..., x p ) dp (i 1,..., i p ) R p j, } 4M 1(2$) ; $(1$) (i j1 &i j ). (6.1) As a special case, if h(x 1,..., x p ) is bouded, say, h(x 1,..., x p ) M, the we ca replace the right side of (6.1) by 2M;(i j1 &i j ). Lemma 6.2. Let [X i ] be strog mixig. Let Z 1 be _(X i,1i j)- measurable (1 j) ad Z 2 be _(X i, i jm)-measurable. If E( Z 1 p ) <, E( Z 2 q )< ad r p q =1(r, p, q>0) the E(Z 1 Z 2 )&E(Z 1 ) E(Z 2 ) 12((m)) 1r [E Z 1 p ] 1p [E Z 2 q ] 1q. (6.2) Let W = 12 (V &V ). For a rectagle I, put W (I)= 12 (~ (I)& (I)), where ~ ad are the probability measures pertaiig to V ad V.
276 HAREL AND PURI Uder the coditios of Theorem 2.3, we have Lemma 6.3. Uder the coditios of Theorem 2.3, let B=[a 1, b 1 ]_ [a 2, b 2 ] ad B$=[c 1, d 1 ]_[c 2, d 2 ] be two rectagles icluded i [0, 1] 2 such that B$/B, the W (B$) W (B) 12 (B). (6.3) Proof. We have W (B$)= 2 &(H i (F H i (F 2 &(H i (F H i (F 2 &H i (F &(H i (F H i (F O the other had { I [c1 F (X i )d 1 ]I [c2 G (Y )d 2 ] (d 1), G (c 1 ), G (d 2 ))H i (F (d 2 ))&H i (F { I [a1 F (X i )b 1 ]I [a2 G (Y )b 2 ] (b 1 ), F (a 1 ), F { H i (F (a 1 ), F (d 1), G (c 1 ), G (b 2 ))H i (F (a 2 ))&H i (F (b 1 ), F (a 2 ))H i (F (d 2 ))H i (F (d 2 ))&H i (F (d 1 ), G (c 2 )) (c 1), G ( 2 ))) = (b 1 ), F (a 2 )) (a 1 ), F (a 2 ))) = (b 2 ))&H i (F (a 1), F (a 2 )) (d 1 ), G (c 2 )) (b 1 ), F (a 2 )) (c 1), G (c 2 ))) = W (B) 12 (B). (6.4) &W (B$)=& 2 &(H i (F H i (F { I [c1 F (X i )d 1 ]I [c2 G (Y i )d 2 ] (d 1 ), G (c 1 ), G (d 2 ))H i (F (d 2 ))&H i (F (d 1 ), G (c 2 )) (c 1 ), G (c 2 ))) = 2 (B$) (B) 12 (B) W (B). (6.5) From (6.4) ad (6.5), we deduce (6.3).
CONDITIONAL EMPIRICAL PROCESSES 277 Lemma 6.4. Uder the coditios of Theorem 2.3 we have E(W (I)) 2q C \ 2l(2$) q &q&l ( (I)) (6.6) l=1 for ay rectagle I of [0, 1] 2 ad $>0. Proof. We will prove it for oly q=2, the proof for q>2 is similar ad is therefore omitted. Deote A i =I [u1 F (X i )u 2 ] } I [v1 G (Y i )v 2 ] &(H i (F &H i (F =B i &C i, (u 2 ), G (u 1 ), G where I=[u 1, u 2 ]_[v 1 v 2 ]. We have E(W (I)) 4 &2 (v 2 ))&H i (F (v 2 ))H i (F E(A \, i A, j A, k A, l ) 1i, j, k, l &2 4! \ &2 &2 =4!(K (1) &i j=1 &i ki li j=1 ki li &i K (2) j=1 ki li The from Lemma 6.2, it follows that &2 K (1) &2 &2 (u 2 ), G (v 1 )) (u 1 ), G (v 1))) E(A, j A, ji A, jik A, jikm ) E(A, j&i&k&l A, j&i&k A, j&i A, j ) E(A, j&l A, j A, ji A, jik ) (3) K ). (6.7) &i i 2 (;(i)) $(2$) i 2 (;(i)) $(2$) i 2 (;(i)) $(2$) j=1 (C j ) 1(2$) (C, ji ) 1(2$) &i \ C j=1 j \ C j=1 j &i \ 1(2$) 1(2$) C, j=1 ji 1(2$) 1(2$)\ C j=1 j
278 HAREL AND PURI Similarly &2 = &2 \ (;(i)) $(2$) i 2 (;(i)) $(2$)\ \ 2(2$) \ C j=1 j 2(2$) C j=1 j i 2 (;(i)) $(2$) ( ([u 1, u 2 ]_[v 1, v 2 ])) 2(2$) C ( (I)) 2(2$). (6.8) From Lemma 6.2, we also obtai K (3) &2 \ &i &2\ \ K (2) C( (I)) 2(2$). (6.9) j=1 ki li &i j=1 C ( (I)) 2(2$) &2 \ &i i 2 (;(i)) $(2$) E(C, j&l C j ) E(C, ji C, jik ) j=1 E(C j C, ji ) &i (;(i)) $(2$) C ( (I)) 2(2$) j=1 (C j ) 1(2$) (C, ji ) 1(2$) &k k=0 l=1 C l C, lk 1(2$) 2 (C j ) 1(2$) (C, ji ) C(( (I)) 4(2$) ( (I)) 2(2$) ). (6.10) From (6.7) through (6.10), we obtai (6.6) ad Lemma 6.4 is proved. ad For ay '>0, ad ay # N, let c 0 >0 be give, put (')=sup [ W (u, v)&w (u$, v$), v&v$ <&, u&f(x 0 ) <c 0 a, u$&f(x 0 ) <c 0 a ]. w *(')=sup [ W (u, v)&w (u, v$)&w (u$, v)w (u$, v$), v&v$ <', u&f(x 0 ) <c 0 a, u$&f(x 0 ) <c 0 a ].
CONDITIONAL EMPIRICAL PROCESSES 279 Lemma 6.5. Uder the coditios of Theorem 2.3, we have lim lim sup P( (')=)=0, \=>0. (6.11a) ' 0 \!>0, \=>0, _'>0,! 0 # N, such that \ 0 P(w *(')=)!(a ) (2&$)(2$). (6.11b) Proof. Let be fixed. We cosider a subdivisio of [0, 1] 2,(u i, v j )# [0, 1] 2,1i, 1j such that u 0 =0<u 1 <}}}<u =1 v 0 =0<v 1 <}}}<v =1 u i &u i =, v j &v j =,,..., j=1,..., ad we cosider also a subdivisio of [0, 1], v$ 0 =0<v$ 1 <}}}<v$ k =1 such that v$ l &v$ l <'2 v$ l &v$ l '2, 1lk v$ l # [v 0, v 1,..., v ]. We defie (a, b)#[0,1] 2 by F(x 0 )&a<c 0 a, (a, b)#[u 0, u 1,..., u ] F(x 0 )&a& c 0 a b&f(x 0 )<c 0 a, b &F(x 0 )c 0 a. For ay l # [1,..., k] ad ay i # [0,..., ] we defie a sequece of r.v.'s Z (l) (i, j)=w (u i, v$ l j )&W (u i, v$ l ), 0i, 1jp, where v$ l p =v$ l. For ay j # [0,..., ] we defie aother sequece of r.v.'s Z (0) (i, j)=w (a(i), v j )&W (a&, v j ), 0im, 1j, where a(m) =.
280 HAREL AND PURI From Lemma 6.4 for q=4, we deduce ad E(Z (l) (i, j)&z (l) (i$, j$)) 8 ( [i$, i ]_[0, j]) 8(2$) ((i&i$) ) 4(2$), where 0i$i, 1lk (6.12) E(Z (0) (i, j)&z (0) (i, j$)) 8 ([0, i]_[ j$, j ]) 8(2$) (( j& j$) ) 4(2$). The by Theorem 12.2 of Billigsley (1968), we have \=>0, \l # [1,..., k], \i # [0,..., m] 4(2$) P[ sup Z (l) (i, j) >=]C \' 0 jp 2 (6.13) ad \=>0, \j # [0,..., ] P[ sup Z (0) (i, j) >=]C(2c 0 a ) 4(2$). (6.14) 0im Defie ow for ay l # [1,..., k] Z * (l) (i, j)=w (a(i), v$ l j )&W (a&, v$ l j ) &W (a(i), v$ l )W (a&, v$ l ). From Lemma 6.4 for q=4, we deduce E(Z * (l) (i, j)&z * (l) (i$, j)&z * (l) (i, j$)z * (l) (i$, j$)) ( ([i$, i ]_[ j$, j ])) 8(2$) ((i&i$) ( j& j$)) 4(2$), where 0i$<im, 0j $< jp. (6.15) The by Lemma 1 of Balacheff ad Dupot (1980) we have \=>0, P[ sup Z (l) (i, j) >=]C(c 0im * 0 a ') 4(2$). (6.16) 0 jp For ay (u i, v j ) ad ay (u i$, v j$ ) such that v j &v j$ <' ad u i, u i$ # [a, a,..., b&, b]; Suppose u i <u i$ ad v$ l v j$ <v j v$ l
CONDITIONAL EMPIRICAL PROCESSES 281 W (u i, v j )&W (u$ i, v$ j ) =W (u i, v j )&W (a, v j ) &W (u i, v$ l )W (a, v$ l )&W (u i$, v j$ )W (a, v j$ ) W (u i$, v$ l )&W (a, v$ l )W (a, v j )W (u i, v$ l ) &W (a, v j$ )&W (u i$, v$ l ). (6.17) If v$ l <v j$ <v$ l v j, we write W (u i, v j )&W (u i$, v j$ ) From (6.17) ad (6.18), we deduce sup v j &v j $ <' F(x 0 )&u i <c 0 a F(x 0 )&u i $ <c 0 a sup 1lk =W (u i, v$ l )W (u i, v j )W (u i, v$ l ) sup 1lk &W (u i, v$ l )W (u i, v j )&W (u i$, v j$ ). (6.18) W (u i, v j )&W (u i$, v j$) sup Z (l) (i, j) 3 sup 0im * 1lk 0 jp sup 0im sup Z (l) (i a, j) 0 jp Z (0) (i, j l ), (6.19) where i a =a ad v jl =v$ l. By usig (6.13), (6.14), ad (6.16), we obtai \=>0 P[ sup W (u i, v j )&W (u i$, v j$) >=] v j &v j $ <'2 F(x 0 )&u i <c 0 a F(x 0)&u i $ <c 0a C {k(c 0a ') 4(2$) 3k 2 4(2$)= 4(2$) \' k(2c 0 a ) \ k' 2\ ' (2&$)(2$) 2 3 \k' 2 \ ' (2&$)(2$) 2 \ ' (2&$)(2$) 2 C {(2c 0a ) 4(2$) C {(2c 0a ) 4(2$) k(2c 0 a ) 4(2$)= 3 2 4(2$)= (2&$)(2$) \' k(2c 0 a ) (6.20)
282 HAREL AND PURI which implies \=>0, \*>0, _'>0, _ 0 # N, \ 0, P[ sup W (u i, v j )&W (u i$, v j$) =]<*. (6.21) v j &v j $ <'2 F(x 0 )&u i <c 0 a F(x 0 )&u i $ <c 0 a Let (u, v) ad (u$, v$) # [0, 1] be such that v&v$ <' ad F(x 0 )&u < c 0 a, F(x 0 )&u$ <c 0 a. There exist (i, i$) # [1, 2,..., ] 2 ad ( j, j$) # [1, 2,..., ] 2 such that We have u i uu i, u i$ u$u i$, v j vv j, v j$ v$v j$. W (u, v)&w (u$, v$) W (u i, v j )&W (u i$, v j$ ) W (u i, v j )&W (u i, v) W (u i, v)&w (u, v) W (u i$, v j$ &W (u i$, v$)) W (u i$, v)&w (u$, v$). By usig Lemma 6.3, we get W (u, v)&w (u$, v$) W (u i, v j )&W (u i$, v j$ ) W (u i, v j )&W (u i, v j ) W (u i, v j )&W (u i, v j ) W (u i$, v j$ )&W (u i$, v j$ ) W (u i$, v j$ )&W (u i$, v j$ ) 4 2 (6.22) From (6.21), we deduce \=>0, \*>0, _'>0, _ 0 # N, \ 0, P[ sup ad (6.11a) is proved. W (u, v)&w (u$, v$) =]<* (6.23) v j &v j $ <'2 F(x 0 )&u i <c 0 a F(x 0 )&u i $ <c 0 a The proof of (6.11b) is similar. Lemma 6.6. We have P[ lim M (x 0 )=x 0 ]=1, 0<x 0 <1. (6.24)
CONDITIONAL EMPIRICAL PROCESSES 283 Proof. From Lemma 6.4 we easily deduce that E(M (x 0 )&x 0 ) 4 C( &3 (x 0 ) 2(2$) &2 (x 0 ) 4(2$) ) C( &3 &2 ). (6.25) By the Markov iequality P[ M (x 0 )&x 0 >=]C &2 = &4 (6.26) ad so by the BorelCatelli lemma P[ M (x 0 )&x 0 = i.o.]=0 (6.27) for each positive =, which implies (6.24). So we have the lemma. Lemma 6.7. Let =>0 be arbitrarily. The there is a costat C, such that, up to a eve of probability less tha or equal to =, we have wheever t&s Ca (6.28) M (t)&m (s) a. (6.29) Proof. From Theorems 6.1 ad 14.3 of Billigsley (1968) ad the fact that the sequece of probability measures P defied by W is tight, we deduce that there exists some fiite umber M such for all sufficietly large, P(A (M))1&=, where A (M)=[sup 0t1 M (t) M 2 ]. From (6.29), o the set A (M) we have t&s t&m (t) M (t)&m (s) M (s)&s = M (t) M (t)&m (s) M (s) ad the proof is completed. a 2M 2 Ca
284 HAREL AND PURI Lemma 6.8. If the sequece [X i ] is strog mixig with rates (2.12), the for ay *>0, P[ sup stsc 1 a M *(t)&m *(s) *a 12 ]C*&4, (6.30) where M *= 12 (M &M ). Proof. From Lemma 6.4, if t&s=, the E M *(t)&m *(s) 4 C= (t&s) 4(2$) for all. Hece followig the method of the proof of (22.20) i Billigsley (1968), we get P[ sup M *(t)&m *(s) *a 12 ]C= * &4 (mp) 4(2$). (6.31) stsmp Now puttig mp=c 1 a, we have (6.30) from (6.31). The followig lemma is Theorem 1 of Yokoyama (1980). Lemma 6.9. Let [Y j ] be a strog mixig sequece with E(Y j )=0, j1, ad sup j1 E Y j r$ < for some r>2 ad $>0. If (i1) (r2) [(i)] $(r$) < (6.32) the E } r Y i} C r2, 1. (6.33) REFERENCES 1. S. Balacheff ad G. Dupot, Normalite asymptotique des processus empiriques troque s et des processus et rag, i ``Lecture Notes i Mathematics,'' Vol. 821, pp. 1945, Spriger-Verlag, New YorkBerli, 1980. 2. G. Beett, Probability iequalities for the sum of idepedet radom variables, J. Amer. Statist. Assoc. 57 (1962), 3345. 3. P. Billigsley, ``Covergece of Probability Measures,'' Wiley, New York, 1968. 4. Yu. A. Davydov, Mixig coditios for Markov chais, Theory Probab. Appl. 18 (1973), 312328. 5. L. P. Devroye ad T. J. Wager, O the L 1 covergece of kerel estimators of regressio fuctios with applicatios i discrimiatio, Z. Wahrsch. Verw. Gebiete 51 (1980), 1525. 6. P. Doukha ad F. Portal, Pricipe d'ivariace faible pour la foctio de re partitio empirique das u cadre multidimesioel et me lageat, Probab. Math. Statist. 8, Fa. 2, (1987), 117132.
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