V7". ii- . ‘\‘h w: ‘ WK ".2! OVERDUE FINES: 25¢ per day per item RETURNING LIBRARY MATERIALS: Place in book return to remove charge from circulation records © 1980 ALFRED JOSEPH VANDERZANDEN All Rights Reserved SOME RESULTS FOR THE WEIGHTED EMPIRICAL PROCESS CONCERNING THE LAW OF THE ITERATED LOGARITHM AND WEAK CONVERGENCE By Alfred Joseph Vanderzanden A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Statistics and Probability 1980 'ABSTRACT SOME RESULTS FOR THE NEIGHTED EMPIRICAL PROCESS CONCERNING THE LAW OF THE ITERATED LOGARITHM AND WEAK CONVERGENCE By Alfred Joseph Vanderzanden In this paper we establish two main results for the weighted empirical process. The first result is a functional law of the iterated logarithm when the underlying random variables are i.i.d. Uniform [0,ll. The second result is the weak convergence of the weighted empirical process to a Gaussian process with almost sure continuous sample paths when the underlying random variables repre- sent an array of row independent random vectors taking values in the k-dimensional unit cube [0,llk. To Joyce ii ACKNOWLEDGEMENTS I wish to express my sincere thanks to Professors Hira Koul and Joel Zinn for their guidance and especially their patience during the preparation of this dissertation. I would also like to thank Professors Roy Erickson and Joel Shapiro for their review of my work. Finally, special thanks go to Mrs. Noralee Burkhardt for her excellent typing of the manuscript. Section 1 2 TABLE OF CONTENTS SUMMARY A FUNCTIONAL LAW OF THE ITERATED LOGARITHM FOR THE WEIGHTED EMPIRICAL PROCESS l. Introduction 2. The Law of the Iterated Logarithm for Weighted Empirical Processes 2. 2. WEAK CONVERGENCE OF THE WEIGHTED EMPIRICAL PROCESS WITH MULTIDIMENSIONAL PARAMETER 3.l. Introduction 3.2. Weak Convergence of the Weighted Empirical Process APPENDIX BIBLIOGRAPHY iv Page TB 18 21 34 4O I. SUMMARY The weighted empirical process deserves special recognition among stochastic processes. It serves as a fundamental tool in the study of statistics based on ranks, as they occur in nonparametric statistics, because of the ability to express these rank statistics in terms of the weighted empirical process (see Koul (1970a) and Koul and Staudte (1972)). Furthermore, some statistical procedures have recently been proposed by Sinha and Sen (1979) and Koul (1980) which involve the weighted empirical process directly. In this paper we establish two main results for the weighted empirical process. The first result is a functional law of the iterated logarithm when the underlying random variables are i.i.d. Uniform [0,1]. This appears in Section 2 as Theorem 2.2.1 and extends the work of Finkelstein (1971) using some ideas feund in James (1975) and Kuelbs (1976). The second result appears in Section 3 as Theorem 3.2.2 and establishes the weak convergence of the weighted empirical process to a Gaussian process with almost sure continuous sample paths when the underlying random variables repre- sent an array of row independent random vectors taking values in the k-dimensional unit cube [0,llk. Theorem 3.2.2 extends the work of Koul (1970), Withers (1975), and Shorack (1980) using the fluctuation inequalities of Bickel and Wichura (1971) which we present in Section 4. 2. A FUNCTIONAL LAW OF THE ITERATED LOGARITHM FOR THE WEIGHTED EMPIRICAL PROCESS . 2.1. Introduction Let Y],Y2,... be.a sequence of random variables defined on a probability space (0,F,P) such that P(Yi 6 [0,1]) = 1 for each i = 1,2,... . Furthermore, let c],c2,... be a sequence of real numbers and define N (2.1.1) as = X c? , N = 1,2,... In this section the weighted empirical process is defined by (2.1.2) VN(t) = N 1: We also define the "normalized" weighted empirical process by (2.1.3) XN(t) = VN(t)/ 2 ofi log log 03 , t e [0.13. Functional laws of the iterated logarithm have been established for the XN process by Finkelstein (1971) when Y],Y2,... are i.i.d. Uniform [0,1] and c1 = l for all i; and also by Philipp (1977) when Y],Y2,... are strictly stationary strongly mixing Uniform [0,1] and Ci = 1 for all i. James (1975) and Wellner (1977) have extended Finkelstein's (1971) result to empirical processes of the form wX where w(t), t 6 [0,1] is a N 'suitable weight function, Y1,Y2,... are i.i.d. Uniform [0,1], 2 and c1 = l for all i. In this paper we extend Finkelstein's (1971) result in a way different from James and Wellner. We still require Y1,Y2,... to be i.i.d. Uniform [0,1] but we allow the weights c],c2,... to be arbitrary real numbers satisfying two regularity conditions. Before stating the main result we introduce some notation adopted from Kuelbs (1976). If (M,d) is a metric space and A ;.M we define the dis- tance from x e M to A by d(x,A) = inf{d(x,y) : y e A}. If {xN} is a sequence of points in M, then C({xN}) denotes the cluster set of {xN}. That is, C({xN}) is the set of all possible limit points of the sequence {xN}. We write {xN} ++ A if both *2: d(xN,A) = 0 and C({xN}) = A. If F(s,t), s,t E T c (-w,m) is a nonnegative definite real- valued function, we define H(F) to be the reproducing kernel Hilbert space generated by the kernel P and “'"H(r) denotes the associated norm on H(r). For an extensive discussion of reproducing kernel Hilbert spaces see Aronszajn (1950). 2.2. The law of the iterated logarithm for weighted empirical processes Let the space of real-valued functions on [0,1] which are right continuous on [0,1] and have left limits on (0,1] be denoted by D[0,l]. Endow the space D[0,lJ with the metric generated by the supremum norm (2.2.1) ”XIIco = sup{|x(t)| : t 6 [0,1]}, x E D[O,1], and let D denote the o-field generated by the H-Hm-open balls of D[0,l]. Theorem 2.2.1. If Y],Y2,... are i.i.d. Uniform [0,1] random vari- ables and c],c2,... are any real numbers satisfying 2 2 _ 2 log log ON _ (2.2.2) lim ON - m and lim ( max c1) 2 - 0, N—roo N-voo 1_<_1'5N ON then with respect to (D[0,l]. D, “'“m) we have (2.2.3) P(XN ++ B) = 1 where XN is the normalized weighted empirical process (2.1.3) and B = {x E H(F) : flqu(r) 5.1} with F(s,t) = (s A t) - st, s,t 6 [0,1]. Theorem 2.2.1 will follow from Lemma 1.1 in Kuelbs (1976) once we establish Lemma 2.2.1 and Lemma 2.2.2 which we now state. Lemma 2.2.1. Suppose the assumptions of Theorem 2.2.1 are satisfied and let T denote any finite subset of [0,1], then with respect to (RT, “-HRT) we have (2.2.4) P(X; ++ 3T) = 1 where RT is the space of all real-valued functions defined on T, "x" = max |x(t)| for all x 6 RT, XT is the process (2.1.3) RT tET N restricted to T, BT = {x e H(FT) : ”x" 1 5_1}, and rT is the restriction of F(s,t) = (s A t) - st ESP I x T. Before stating Lemma 2.2.2 we introduce some additional nota- tion adopted from Kuelbs (1976). If T = {t0,t1,...,tm} where 0 = t0 < t1 <...< tm = 1 and if x E D[0,l]. then we define AT(x) to be the continuous polygonal function such that ti-t t-tM (2-2-5) AT(X)(t) = (t;_:_t;:;)x(ti-l) + (E;f:ff;:;)x(ti) for t 6 [ti-1’ ti] and i l,2,...,m. Lemma 2.2.2. Suppose the assumptions of Theorem 2.2.1 are satisfied, T = {tO’tI’°°"t } where 0 = to < t] <...< tm = I, and m T max lti ' ti"' 5?! lgigm (2.2.6) P(llXN - AT(XN)“m'5-¢T for all sufficiently large N) = l _ 80(‘1 ' t1-1) where XN is the process (2.1.3), “T - 2 max 1 _ (t1 _ t1-17 . and p > 1. Lemmas 2.2.1 and 2.2.2 will be proved shortly, but we first show they imply Theorem 2.2.1. With this in mind let T M = 1,2,... M’ denote any sequence of increasing finite subsets of [0,1] such that the points in TM satisfy < t <...< t for M = 1,2,... M1 M ’mM (2.2.7) | | and lim max t . - t . = 0. If the assumptions of Theorem 2.2.1 are satisfied, then Lemma 2.2.2 gives P("XN - AT (XN)”w'5-9T for all sufficiently large N) = l M M ____ for each M = 1,2,... Hence, it follows that P(lim “XN - AT (XN)”0° < New M q? for each M) = 1. Using (2.2.7) we have lim YT = 0 so that M Wha> M 2.2.8) PT'_T’_IX -A (X) =0)=1. ( ($2.13,“ T”NIL, Relation (2.2.8) shows that condition (ii.c) of Lemma 1.1 in Kuelbs (1976) is satisfied. Since P(s,t) = (s A t) - st is continuous on [0,1] x [0,1], Lemma 3 in Oodaira (1972) implies that B = {x E H(r) : "x”H(P) 5_1} is compact in (C[0,l]. "-fl») where C[0,1J is the space of all real-valued continuous functions on [0,1] and "x“0° = sup{|x(t)| : t 6 [0,1]} for all x E C[0,1]. Furthermore, from the Theorem on page 351 in Aronszajn (1950) it is easy to show that (2.2.9) {x 6 H(PT) : "x" T 5_1} = {x = yT : y e H(F) and H(P ) IM'H(I‘)— < 1}: where P can be any nonnegative definite function on [0,1] x [0,1] and T any finite subset of [0,1]. Hence, Lemma 2.2.1 applied to each TM’ M = 1,2,... shows that condition (i.) of Lemma 1.1 in Kuelbs (1976) is satisfied. Thus, Theorem 2.2.1 now follows from Lemma 1.1 in Kuelbs (1976). Proof of Lemma 2.2.1. Let T = {t],t2,...,tm} denote any finite subset of [0,1]. Let Y],Y2,. .. and c1,c2,... be as in Theorem 2.2.1 and define Z. 1 (Zi(t]), Zi(t2)""’ Zi(tm))’ i = 1,2,... where Zi(t) cJ[I( i < t) - P(Y i < t)], t 6 [0,1], i = 1,2,... N T 2 2 Then XN = (XN(t]), XN(t2),...,XN(tm)) = i2] Zi4J/2 ON log log ON :21: Cov(Z1.)=(I‘(ti,tj))1 j- _] =PT where P(s,t) = (s A t) - st. ON Lemma 2.2.1 now follows immediately from the multivariate law of the iterated logarithm, Thereom 1 in Berning (1979), applied to Zi’ 1 = 1,2,... U The proof of Lemma 2.2.2 depends on several results which we present in the following Lemmas. Lemma 2.2.3. If T = {t0,t],...,tm} where 0 = t0 < t1 <...< tm = l and X(t), t 6 [0,1] is any stochastic process, then (2.2.10) "x - AT(X)un.g, max [ sup |X(t) - X(ti_1)l + 151:1“ tGECT'] ,tiJ sup |X(ti) - X(t)|] tett1_1.til where AT(X) is the continuous polygonal function defined in (2.2.5). Proof. For any t 6 [0,1] there exists 1 e {l,2,...,m} such that t1._1 5_t_g ti' Hence, t1 ' t t ' t1_'| |X(t) - 17mm = l—ti - t“ Wt) - mm“ + ———t. - t. [x(t) - X 0 we have (2.2.20) E exp(aX) §_exp{a2f(aB)Var(X)} ' where (2.2.21) 8 = max [Ic I max{1, P(Y. e (a,b])/P(Y. t (a,b])}] ISAEN 1 1 1 and f(x), x E (~w5m) is the positive, strictly increasing, continuous function defined by (2.2.22) (eX - 1 - x)/x2 if x f o f(X) = 1/2 if x = 0 . 3399:. For each i = l,2,...,N let X, = [Ci/P(Yi t (a,bl)][I(Yi e (a,b]) - P(Yi e (a.b])1. Then it is clear that X1,X2,...,XN are independent random variables with EXi = 0 and |Xi| 5_B, i = l,2,...,N ‘where B is defined in (2.2 21). With f as defined in (2.2 22) we have ex = 1 + x + x2f(x) so that for all a > 0 and i = l,2,...,N we have E exp(aXi) E[l + aXi + azxg 2 f(axi)] 1 + a Ex§f(axi) 1 + a2f(a8) Ex? IA §_exp{a2f(a8) Var(Xi)} since 1 + x 5_ex. 11 Therefore, N E exp(aX) = E exp(a )1 xi ) i- N = H E exp(aXi) i=1 N 2 .3 H exp{d f(aB) Var(Xi)} i=1 = exp{a2f(a8) Var(X)}. U Lemma 2.2.6. Suppose A(N), N = 1,2,... is a nondecreasing sequence of positive numbers and {UN(t), t e T}, N = 1,2,... is a sequence of independent stochastic processes defined on a probability space (0,F,P) and taking values in the space of real-valued functions de- fined on T.: (dngn). Define {WN(t), t E T} by WN(t) = 1:] Ui(t) and assume there is a countable subset {tj, j = 1,2,...} of T such that sup{|WN(t)|, t 6 T} = sup{|WN(tj)|, J = 1,2,...}. Then for any positive integers N1 5_N2 and 6.: w(N],N2) we have ( ) ( INN (t)| '“N2(t)' E A1N])) 2.2.23 P max sup < 2P(sup >-— N1 0. Since A(N), N = 1,2,... is a nondecreas- ing sequence of positive numbers, the event { IWN ml max sup N15N5N2 tGT A)”; 12 is contained in the event A where > 11 f max sup |WN(t)| > e A(N1)} N15N§N2 tET (2.2.25) { _max sup |WN(tj)| > e A(N1)} N15N5N2 J3] For N e {N1,...,N2} and j 6 {1,2,...} define (2.2.26) BN’ = { max max lwn(ti)l-5 e A(N]) and J ngngN lgi e A(N])1 and (2.2.27) ch = {IWN2(tj) - WN(tj)| §_%-e A(N])}. It is clear that for each N E {N}....,N2} the family {BNj’ j = 1,2,...} consists of pairwise disjoint events. Further- more, since BNj depends only on {Ui(t), t 6 T}. i = l,2,...,N and CNj depends only on {Ui(t), t 6 T}, i = N+l,...,N2, we have for each N 6 {N1,...,N2} that the families {BNJ’ j = 1,2,...} and {CNj’ j = 1,2,...} are statistically independent since the processes U],U2,... are independent. Extending Loéve's Lemma for Events on page 246 in Loéve (1963) to countable collections of events we obtain N 2 00 (2.2.28) [inf{P(CNj), N1 §_N 5_N2, j 3_1}]P( u ,g BNj).g . N-N] J'] N2 m P(U 0 B.£ ). N=N1 i=1 NJ NJ N It is easy to show A = u E uT‘ B . where A is defined . . 2 co 1n (2.2.25). 0n the other hand, in the event UN=N] Uj=1 BNchj 13 we have for some N E {N],...,N2} and some j 6 {1,2,...} that 1 |WN(tj)| > e A(N]) and |WN2(tj) - WN(tj)| 5_§-e A(N]). Therefore, 1 INN2(tj)l >-§ e A(N]) and it follows that N2 on ("N 2(t)| E MNl) (2.2.29) 0 11 B .c .c {sup } . N=N1 i=1 N3 N3 tGT A(N27 ”I 2) Furthermore, for each N 6 {N}....,N2} and j 6 {1,2,...} Chebysev's inequality gives 1 - p(ch) = p(|wN2(tj) - wN(tj)| > §-A(N1)) _<_ (FA—imfiwuwnzuj) - wNujn. Using the definition of WN and the independence of UN’ N = 1,2,... we get Var(WN2(tj) - WN(tj)) 5_Var(WN2(tj)) - Var(WN](tj)). There- fore, (2.2.30) sup [1 - P(CN 51)] < sup (mm) ZL-NVBYTW 2(t)) - Var(WN 1(t))]. j>l N1 sup 28 [Var(WN (1)) - Var(WN (t))]. 2 1 t€T A (N1) Hence, if (2.2.31) is satisfied, then (2.2.25), (2.2.28), (2.2.29), and (2.2.30) give INN 2(t)| M(N ) NA) 5. 2 ”:21; W) .2.€_(_N_1_)_) and the lemma is proved. D 14 Proof of Lemma 2.2.2. Let 0 g_a < b g_1 such that b-a 5_%-. Recall that c1,c2,... is a sequence of real numbers such that a: = {N_ _1 c2 +.m and log log ON 2 2 ( max c1) 2 +0 as N + w. Hence, cN/UN-l + 0 as N + m. l 1, set A(N) fu/E a: log log ON’ and choose a positive integer N0 such that A(N) > 0 and cfi/o§_1 5_p - 1 for all N 3.N0. Next choose a number n such that as 5 p". Finally, for each 0 k = 1,2,... define (2.2.32) N(k) = min{N_>_.No : 0N pN+k}. It is easy to show that the sequence {N(k), k = 1,2,...} has the following properties: (2.2.33) N(k) < N(k+1) and pNTk < afi(k) g p"+k+‘ for k = 1,2,. . (2 2 34) (im .JNthll _ p - 11m x :(N(k+1)), kem °N(k) k+w A 2(N(k)) Let VN(t) = 2N=1 ci[I(Yi 5_t) - P(Yi 5_t)] be the weighted empirical process (2.1.2) where Y],Y2,... are i.i.d. Uniform [0,1] random variables. Furthermore, let {WN(t), t 6 [a,b]} denote either {VN(t) - VN(a), t 6 [a,b]} or {VN(b) - VN(t), t 6 [a,b]}. We now apply Lemma 2.2.6 to {WN(t), t 6 [a,b]}, A(N) fiV/2 ON log log 0N , = N(k) and N2 = N(k+1) for k = 1,2,... to Obtain 15 IWN(t)| (2 2.35) P( max sup > e) N(k)§N§N(k+l) teta.bJ A)"; E 3.2 P(tzgg,b] INN(k+])(t)| > fi'A(N(k))) provided 8 «- _.t€[:,b] A2(N(k)) [var(wN(k+])(t)) Var(WN(k)(t))]. Using the fact that Y],Y2,... are i.i.d. Uniform [0,1], it is easy to see that the right-hand side of (2.2.36) is less than or equal to 8Lo§(k+1) - o§(k)3(b-a)/x2(u(k)). Hence, (2.2.35) will hold if 2 2 2 2 Using Lemma 2.2.4 we obtain that VN(t) - VN(a) MN(t) = 1 _ (t-a) , t E [a,b] is a martingale and VN(b) “ VN(t) ' . . RN(t) = 1 _ (b-t)’ , t 6 [a,b] 15 a reversed martingale. In the case "N(t) = VN(t) - VN(a) we have for all a > 0 and 6 > 0 P( sup IWN(t)| > 6).: P( sup IMN(t)| > a) t€[a.b] tE[a,b] 5.P( sup MN(t) > a) + P( sup (-MN(t)) > a) t6[a.b] t6[a,b] (2.2.38) P( sup exp(aMN(t)) > exp(a6)) tELa,b] + P( sup exp(-aMN(t)) > exp(a6)) teta.b] _g exp(-a6)[E exp(aMN(b)) + E exp(-aMN(b))]. 16 Inequality (2.2.38) follows from Theorem 3.2, page 353 in DOob (1953) since exp(aMN(t)) is a submartingale for all a E (-w,w). Applying Lemma 2.2.5 with x = MN(b) and x = -MN(b) (2.2.33) can be continued to yield (2.2.39) P( sup INN(t)|:>6) g 2 exp{-a6 + a2f(aBN)Var(MN(b))} tEEa.b] where (2.2.40) 8 = max c. N lgjgN I " and Var(MN(b)) = a§(b-a)/ri - (b-a)]. Hence, (2.2.35), (2.2.37) and (2.2.39) give P( INN(t)l ) , max sup -—1-7—-> e N(k)5N5N(k+l)t€[a.bJ A N (2.2.41) - aeA(N(k)) 2 2 b-a 5-4 exP{ 2 + “ f(aBN(k+l))°N(k+l) T’i—TETET} for k = 1 2 a > o and 82 > 8(h-a)Lo2 - 02 J/A2(N(k)) ’ "°" ’ —- N(k+1) N(k) where NN(t) = VN(t) - VN(a). In the same way, (2.2.41) can be shown to hold if NN(t) = VN(b) - VN(a). Now set (2.2.42) a = Aéflifiilll 5 ‘ g_§b‘a) in (2.2.41) ON(k+1) 2’5 and define B A(N(k+1)) - - (2°2'43) Yk = {Eh(é¥i))l ' f‘ N(kgl) SJ— 1 b-§b aJ)‘ “N(k+1) p 17‘ Then the right-hand expression in (2.2.41) can be written as 2 2 (2.2.44) 4 exp{- A2(N(k+l)) fisL-Bjé-fllvk} . ON(k+1) The assumptions imposed on the sequence c1,c2,... and (2.2.34) . a l_ 2 _ 2 2 = if 52 > $E§9i§%37, then for all sufficiently large k we have e2 > 8(b-a)[o§(k+]) - e§(k)J/Az(u(k)) and (2.2.44) is less than or equal to (2.2.45) 4 exp{-A2(N(k+l))/Ufi(k+1)} = 4[log o§(k+])]-2 §_4[log p”+k+‘i'2 = 4[(n+k+l)log pJ'z . Since the series [i=1 4[(n+k+l)log 03-2 < m, the Borel- Cantelli lemma, (2.2.41), (2.2.44) and (2.2.45) give INN(t)I (2.2.46) 0 -_X(N)—'> e for infinitely many k) P( max sup N(k)5N§N(k+l) t€[a,b] IWN(t)| P( sup -——1—7—-> e for infinitely many N). tEEa.b] A N Therefore, for each p > 1, 0 5_a < b 5_l, b-a 5_%- and 2 8 b-a e > 1 _ b-a we have (2 2 47) P( lw"(t)l f 11 ff' i t1 1 N) l . . sup 5,8 or a su 1c en y arge = t6[a.bl A)", where HN(t) is either VN(t) - VN(a) or VN(b) - VN(t). Lemma 2.2.2 now follows from (2.2.47) and Lemma 2.2.3. D 3. WEAK CONVERGENCE OF THE WEIGHTED EMPIRICAL PROCESS WITH MULTIDIMENSIONAL PARAMETER 3.1. (Introduction For each N = 1,2,... let CNi’ i = l,2,...,N be any real numbers and let YNi = (YNil’YNi2""’YNik)’ i = l,2,...,N be k- variate (k.: 1) random vectors taking values in the k-dimensional unit cube [0,llk. In this section we define the weighted empirical process by N - k (3.1.1) VN(t) - igl cNi[I(YNi 5_t) - P(YNi 5.t)], t E [0.1] where, as usual, if x = (x],x2,...,xk) and y = (y],y2,...,yk), then we write x j'y if and only if xi 3y1 for all i = l,2,...,k. We also define the "normalized" weighted empirical process by (3.1.2) ZN(t) = VN(t)/0N , t e [0,13" where ' N 2 _ -2 (301-3) ON - .2 CN'I o l 1 Our goal is to establish sufficient conditions for the ZN process (3.1.2) to converge weakly in the generalized Skorohod metric space (Dk,d) as defined in Bickel and Wichura (1971). To be sure, weak convergence of the ZN process has been studied by many authors under a variety of conditions. Therefore, so that our result can be 18 19 put in perspective with other established results, we shall briefly indicate what has already been done. To begin with, when k = l, (lTIEN cfii)/o§ + 0, and for each N = 1,2,..., YNi’ i = l,2,...,N are EtEtistically independent, Koul (1969), Koul (1970b), Withers (1975), and Shorack (1980) each prove that ZN converges weakly in (Dk,d). Conditions imposed by these authors on the distribution funtions of the YNi vary, but the least restrictive condition is stated in Withers (1975) and Shorack (1980), namely N (3.1.4) lim Tim' sup 17- ) cfii P(YNi E (t,t+6]) = 0. 6+0 N+w t€[0.1—6]oN i=1 Shorack (1973) also proves ZN converges weakly, but is limited to the case cNi = l and an assumption much stronger than (3.1.4) is imposed. Several authors have studied the weak convergence of 2N when YNi’ i = l,2,...,N are not independent but satisfy specific "mixing" conditions. For example, when k = 1 see Billingsley (1968), Sen (1971), Dec (1973), Yokoyama (1973), Yoshihara (1974), Withers (1975), Mehra and Rao (l975), and Koul (1977). The first five authors only consider the case cNi = l and assume YNi’ i = 1,2,... are identically distributed with a continuous distribution function or with a Uniform [0,l] distribution. Withers (1975) and Koul (1977) both assume (3.1.5) sup N( max cfiiwofi < e. N3] 1§J§N Furthermore, Withers (1975) assumes (3.1.4) holds along with some other b 20 regularity conditions. Koul (1977) assumes the average of the dis- tributions P(YNi-i t), i = l,2,...,N is Uniform [0,1]. Mehra and Rao (l975) assume each YNi is Uniform [0,1] and either ( max C§°)/°§ + 0 or sup N5( max cfii)/o§ < m for some 6 > 0 igign ‘ N_>_l 1_<_i_<_N depending on the kind of "mixing" condition assumed. Weak convergence of ZN with respect to metrics stronger than the usual Skorohod metric d has been studied in the case k = l by Pyke and Shorack (1968), O'Reilly (1974), Mehra and Rao (1975), Withers (1976), and Shorack (1980). These authors require cNi = l or (3.1.5) except when YNi’ i = l,2,...,N are either i.i.d. Uniform [0,1] or identically distributed as Uniform [0,1] and satisfy a certain kind of "mixing" condition in which case ( max cfii)/o§ + 0 suffices. lgjgN Among those authors who have studied the weak convergence of 2N when k 3_2 we have Bickel and Wichura (1971), Neuhaus (1971), Sen (l974), RUschendorf (l974), Neuhaus (1975), Yoshihara (1975/76), and RUschendorf (1976). The first six authors each limited their study to the case CNi = l. Bickel and Wichura (1971) also assumed YNi’ i = 1,2,... were i.i.d. with a continuous distribution function while Neuhaus (1971) assumed YNi’ i = 1,2,... were i.i.d. with a distribution function satisfying a Lipschitz condition. Sen (1974) and RUschendorf (1974) both assumed YNi’ i = 1,2,... satisfied a certain "mixing" condition. Sen (1974) also assumed YNi’ i = 1,2,... were identically distributed and had Uniform [0,1] marginal dis- tributions while RUschendorf (1974) assumed 21 sup sup P(YNi E A) 5_u(A) N31 lgigN k for some measure u on [0,1] with continuous marginals. . = l, assumed Neuhaus (1975), in addition to assuming cN1 YNi’ i = 1,2,... were independent and the average of the distribu- tion functions of YNi’ i = l,...,N had Uniform [0,1] marginals. Yoshihara (1975/76) assumed YNi’ i = 1,2,... satisfied a certain "mixing" condition and were stationary in addition to assuming cNi Rilschendorf (1976) is the only author that has studied the weak convergence of the multiparameter weighted empirical process with general weights cNi' Most of the results obtained by Rfischendorf (1976) depend on his Lemma 2.1 appearing on page 913 in the same article. From RUschendorf's description of the proof of Lemma 2.1 it appears to this writer that the proof is incorrect. Hence, at this time no further comment will be made concerning the results in RUschendorf (1976). In this paper we extend the results of Koul (1970b), Withers (1975), and Shorack (1980) to the multidimensional parameter weighted empirical process in the case where YNi’ i = l,2,...,N are statistically independent (see Theorems 3.2.1 and 3.2.2). 3.2. Weak Convergence of the Weighted Empirical Process For k = 1,2,... let (Dk’d) denote the (separable) metric space of real-valued functions defined on [0,1]k which are "con- tinuous from above, with limits from below" as defined in Bickel and Wichura (1971). Furthermore, let Ck denote the set of all continuous 22 real-valued functions defined on [0,1]k k . If t = (t],t2,...,tk) is a point in R we define the norm of t by “t" = max{ltjl : j = l,2,...,k}. Also, if 6 > 0 and x e Dk’ then we define w6(x) to be the usual modulus of continuity, namely k (3.2.1) w5(x) = sup{|x(s) - x(t)| : s,t e [0.1] and us - tfllg a}. The main results of this section concern the "normalized" weighted empirical process (3.1.2) and are stated as Theorem 3.2.1 and Theorem 3.2.2 . k denote the process in (3.1.2) Theorem 3.2.1. Let ZN(t), t 6 [0,1] and assume (3.2.2) YNl’YN2"°°’YNN are statistically independent for each N = 1,2,..., . 2 2 _ (3.2.3) 11m ( max cNi)/°N - 0 , N+w lgjgN and for each j = l,2,...,k -——— 1 N 2 (3.2.4) lim lim sup -§- cNi P(x < YNi' 5_x + 6) = 0 . 6+0 N-mo x€[0,l] 0N i=1 3 Then for all e > 0 (3.2.5) 1im'Tfim P(w6(zN).3 e) = o . 6+0 N+m Theorem 3.2.2. If in addition to the assumptions of Theorem 3.2.1 we also have (3.2.6) lim Cov(ZN(s), ZN(t)) = F(s,t) N+w for all s,t e [0,1]“, then 23 ZN converges weakly in (Dk,d) to a zero mean Gaussian process Z having covariance F and P(Z E Ck) = l. The proof of Theorem 3.2.1 is quite similar to the proof of Theorem 2.2 in Koul (1970b). The main tools used in proving Theorem 3.2.1 are the fluctuation inequalities of Bickel and Wichura (1971) [see Lemma 4.1 and Theorem 4.1 in Section 4 of this paper] while Koul (1970b) uses the fluctuation inequalities in Billingsley (1968). Theorem 3.2.1 will be proved after first establishing four lemmas. The first lemma, Lemma 3.2.1, provides a necessary and sufficient condition for (3.2.5) to hold. This condition (3.2.9) is more convenient to work with than (3.2.5) when applying Bickel and Wichura's (1971) fluctuation inequalities. The second lemma, Lemma 3.2.2, provides sufficient moment inequalities to justify the use of the fluctuation inequalities in Bickel and Wichura (1971). Finally, the third and fourth lemmas, Lemma 3.2.3 and Lemma 3.2.4, provide inequalities from which Theorem 3.2.1 will follow easily. Theorem 3.2.2 will then follow from Theorem 3.2.1 and an easy application of the multivariate version of the Lindeberg-Feller Central Limit Theorem. Keeping the preceding remarks in mind let us for each Borel k set A in R define N -1. (3.2.7) ZN(A) - ON iél cNi[I(YNi e A) - P(YNi e A)]. Furthermore, for each 6 > 0 and j = l,2,...,k let A(j,6) denote the class of all subsets A : [0,1]k having the following form 24 (3.2.8) A = [0,t]] X...x [0,tj_1] x (x,y] x [O’tj+l] x...x [0,tk] where 0 5_y-x g 6. k Lemma 3.2.1. Let ZN(t), t 6 [0,1] denote the process in (3.1.2). Then (3.2.5) lim 1—11TI P(w6(ZN) 3 e) = 0 for all e: > 0 6+0 N+00 if and only if (3.2.9) liml—im P(wé‘j)(ZN) 3 c) = 0 for all e > 0 and j= l,2,...,k 6+0 N*” where (3.2.10) w§5)(zN) = sup{|2N(A)| : A e A(J.6)}, ZN(A) is defined in (3.2.7) and A(j,6) is the class of sets of the form (3.2.8). Proof. First observe that if s = (51’52""’5k) and k with “s - t“.$ 6 t = (t],t2,...,tk) are any two points in [0,1] and u = (5] v t],...,sk v tk), then s 5_u, t §.u, "s - u" 5_6, and “t — u" 5_6. Hence, by the triangle inequality we have |ZN(s) - ZN(t)| 5_|ZN(s) - ZN(u)| + |ZN(t) - ZN(u)| and it follows that (3.2.11) wé(ZN) g "6(ZN) 5_2 wé(ZN) where (3.2.12) 143a") = sup{|ZN(s) - ZN(t)| : s,t e [0,13". s 5 t, H5 - t“ 5.6}. 25 It is now clear that (3.2.5) will hold if and only if (3.2.13) lim Tim P(wé(ZN) 3 e) = 0 for all e > 0. 6+0 N+0° We now show (3.2.13) is equivalent to (3.2.9). Let s = (s1....,sk) and t = (t1....,tk) be any points in [0,1]k with 5.: t and “s - tfl_g 6. Then (3.1.2) and (3.2.7) give (3.2.14) ZN(t) - ZN(s) = ZN([0,tJ \ [0,5]). It is also clear from (3.2.7) that if A and B are disjoint Borel k sets in R , then ZN(A u 8) = ZN(A) + ZN(B). Hence, if we define Aj, j = l,2,...,k by (3.2.15) Aj = [0,5]] x...x [0,sj_]] x (sj,tjl x [0,tj+]] x...x [0,tk], then A],A2,...,Ak is a partition of the set [0,t] \ [0,53 and it follows from (3.2.14) that k (3.2.16) ZN(t) - ZN(s) = 321 ZN(Aj) . Since Aj 6 A(j,6), j = l,2,...,k (3.2.16) gives - . " (3) (3.2.17) w5(ZN) 5_j§1 w6 (ZN). 0n the other hand if A 6 A(j,6), then A has the form A = [0,u]] X...x [0,uj_]] x (x,y] x [0,uj+]] x...x [0,uk] where 0.5_y-x 5_6. 26 If s = (51’52""’Sk) and t = (t].t2,...,tk) are defined by ’ ' = = ° ° k sj - x, tj - y and 5i ”i ti for 1 # J, then s,t 6 [0,1] 9 s 5_t, "s - tug: 6, and A = [0,t] \ [0,5]. Hence, ZN(A) = ZN([0,tJ \ [0,5]) = ZN(t) - ZN(s) and it follows that (3.2.18) wéj)(ZN) 5,wé(zN) for a11 j = l,2,...,k . Lemma 3.2.1 now follows from (3.2.13), (3.2.17), and (3.2.18). D Lemma 3.2.2. Let Y],Y2,...,YN be statistically independent k- variate random vectors taking values in Rk, let c],c2,...,cN be any real numbers, and for each Borel set A in Rk define N ZN(A) = 1;] ci[I(Yi 6 A) - P(Yi E A)] and N 2 uN(A) = .z Ci P(Yi E A) . 1-l Then (3.2.19) E|ZN(A)|2|ZN(B)|2_5 3 n§(A u B) if A and B are disjoint; and (3.2.20) E|ZN(A)|4 5_3 h§(A) + ( max c§)aN(A). lgjgN 2599:. For each Borel set A in Rk and i = l,2,...,N define X(Ai) = I(Yi E A) - P(Yi E A) and P(Ai) = P(Yi e A) . 27 Now let A and 8 denote any two Borel sets in Rk. Since Y1,Y2,..., are independent and E X(A1) = 0 for all i = l,2,...,N and all Borel sets A, we have EIZ (Allzll (B)!2 N N N N ELX(A )X(A )X(B )X(B )] = C.jCCC . . N N 1_1 j- _1 k- 1 £_1 1 k z 1 j k t N (3 2.21) = Z c? EX2(A1)X2(B1) + N N c12c2[EX2(A1)][EX2(B.)] i=1 i= -1 j= —1 c3 J #3 +2 N 2 c2 A1 11N1 321 c1c1LE X( )X(B1)J[E X(Aj)X(Bj)]. #3 Since |X(A1)| g 1 and E x2(A1) §_P(A1), (3.2.21) gives the following result when A = B. N N N cge x4(A1) + 3 N N cfcztez x (A1)J[E x2 (A. )1 i=1 i=1 i=1 J 123 EIZN(A)I4 5_( max c1) N c “P(A + 3[ N c. “P(A )]2 l 0, and x 6 [0,1] let A(j,6,x) denote the class of all subsets A of the form A = [0,t1] x...x [0,t1_1] x (x,t1] x [0,t1+1] X...x [0,tk] where t. 6 Ex, x+6] and t1 6 [0,l] for i f j. Then for each s > 0, J 6 > 0, j = l,2,...,k, and x 6 [0,l] (3.2.22) P(sun{|ZN(A)l: A 6 AU 6,x)} > e) < P(IZN11- (,(x x+ E) 2 3ka(2, 4) +"Y———"Zf'UNj((X:X+5]) where Ck(2,4) is a constant depending only on k. Proof. Let a > 0, 6 > 0, j 6 {l,2,...,k}, and x e [0,l]. For each t = (t1,...,tk) in Rk define N ZN(t) = 1211c1[I(Y1< t) - P(Y1t)]J, tx = (t1,. ,t1_ _1,x, t1+1,.. . ,tk), and (3.2.23) XN(t)= Z N(t)- ZN(tx) . Let us first observe that the fluctuation inequalities in Bickel and Wichura (197l) (see Section 4, Lemma 4.1 and Theorem 4.l in this paper) can be applied to the stochastic process XN(t), t e T where T = T] x T2 X...x Tk, 29 T. J Ex, x+6J , and Ti = [-6, l + 6] for i f j. If t = (t1....,tk) is a point in the lower boundary of T, 2(T) (see Section 4 (4.2)). then either tj = x or ti = -6 for some i f j. If tj = x, then t = tx and XN(t) = 0. If ti = -6, then XN(t) = 0 since v,,...,vN take values in [0,13k. Therefore, XN(t) = 0 for all t E 2(T) so that Lemma 4.l in Section 4 can be applied to give (3 2 24) :2? IXN(t)|.: IXN(b)| + k Iggfik M3(XN) ‘ where b = (b1....,bk), bj = x + 6, and bi = l + 6 for i f j. Since Y],Y2,...,YN take values in [O,l]k, it is easy to see from (3.2.23) that (3.2.25) sup |XH(t)| = sup{IXN(t)| : t e T n [0.13k} tET ' ’ and II IIMZ (3-2-26) XN(b) cl[I(Yij E (X, X + 5]) ‘ P(Ylj 6 (X9 X + 53)] i l ZNj ((x, x + 6]) . Furthermore, if t = (t1....,tk) is a point in T n [0,ljk, then (3.2.23) gives N (3.2.27) XN(t) = ZN(A) = .2] ci[I(Yi e A) - P(Y1 e A)] 1: where A = [0,t13 x...x [O’tj-l] x (x,tj] x [0,tj+]] X...x [0,tk]. Combining (3.2.25) and (3.2.27) yields (3.2.28) :2? |XN(t)| = sup{|ZN(A)| : A 6 A(j.6,x)} 30 If B = (s,tJ n T is a block in T (see (4.3) in Section 4) then by Lemma 4.2, Section 4 the increment (see (4.7) in Section 4) N of ZN(t), t e T around 8 is ZN(B) = ’Xl ci[I(Yi e B) - P(Yi E B)]. 1: Furthermore, the increment of ZN(tx), t e T around 8 is zero. Hence, the increment of XN(t), t E T around 8 is N (3.2.29) XN(B) = ZN(B) = z ci[I(Yi e B) - P(Y1 e 3)] 1 l and Lemma 3.2.2 gives (3.2.30) P(min{|XN(A)|,|XN(B)|} 3_x) = P(min{|ZN(A)|,|ZN(B)|}.3 x) 23;; EIZN(A)|2|ZN(B)IZ 53—4 ufim u a) for all A > 0 and every pair A, B of disjoint neighboring blocks N in T where uN(A) = .2] c? 1: now be applied to yield P(Yi e A). Theorem 4.l in Section 4 can 3ka(2,4) 2 (3.2.3l) P( max M%(XN) z-A)-5'_—__7T__'HN(T \ 2(T)) 1:353 J A for all A > 0 where Ck(2,4) is a constant defined in (4.l3) in Section 4. Since Y],Y2,...,YN take values in [0,l3k, it is easy to see that IIMZ (3.2.32) pN(T \ 2(T)) = c? P(Yij e (x, x + 6]) = uNJ((x, x + 5]). i l Finally, Lemma 3.2.3 follows from (3.2.24), (3.2.26), (3.2.28), (3.2.31), and (3.2.32). D Lemma 3.2.4. Suppose the assumptions of Lemma 3.2.3 hold. Then for each s > O, 6 e (0,l), and j = l,2,...,k 31 (3.2.33) P(sup{|ZN(A)| : A 6 A(j.6)}_: 5).: 2 i N .((x, x + 6]) + 64 max C?) X c l --[d sup n 4 k NJ ljigfl 1 i=l c x€[0,l] where dk is a constant depending only on k. “3592:. Let c > 0, 5 e (0,l), and j E {l,2,...,k}. Further- more, define m(6) = min{m E {l,2,...} : l 5_m6}. For each k t = (t1....,tk) in R define IIMZ ZN(t) = c-[I(Yi §_t) - P(Yi_g t)] and t = (t1....,tj_],x,t. x ..,tk) for x e R. If A(j,6) is the class of sets defined in (3.2.8) and A 6 A(j.6), then A : [0,l]k and has the form A = [0,t]] X...X [0,tj_]] X (x,y] X [0,tj+]] X...X [0,tk] where 0 5_y-x §_6. Clearly, x 6 [m]6, (m1 + l)6] and y 6 [m25, (m2 + l)6] for some integers m],m2 6 {0,l,...,m(6) - l} satisfying either m] = m2 or m2 = m1 + 1. Hence, (3.2.34) |ZN(A)| IZN(ty) - ZN(tx)l IA IZN(ty) - ZN(tm25)l + IZN(tm25) - zN(tm]5)l + IZN(tm]6) - ZN(tx)I 5_3 max sup{|ZN(B)| : B e A(j,6,i6)} Qgi_%-e) < Qgi e).:-lz Edk sup “NJ((X’X+53) + ‘S‘SNZ J XEEO, l] 0N for each N = 1,2,..., j = l,2,...,k, 6 E (0,l), c > 0 where wéj)(ZN) is defined by (3.2.10) in Lemma 3.2.l, dk is a constant depending only on k, ofi = if] cfii, and uNj((x,x+6]) =-—§ 1.Zlcwziflx < YNij 5_x+6). = 0' = N 33 Theorem 3.2.l now follows from (3.2.3), (3.2.4), and Lemma 3.2.1. B Proof of Theorem 3.2.2. By Theorem 2, page 683 in Wichura (1969), the result in Theorem 3.2.2 will follow if each of the following two conditions hold: (i) for each finite subset T of [0,13k, the distribution of ZN(t), t E T converges weakly to a multivariate Normal dis- tribution, and (ii) for each s > 0, lim Tia P(w (z ) > c) = o. 6 N -— 5+0 N+w Condition (ii) holds as a result of Theorem 3.2.1 while con- dition (i) follows from (3.2.6), (3.2.3) and an easy application of the multivariate version of the Lindeberg-Feller Central Limit Theorem. D 4. APPENDIX The paper by Bickel and Wichura (l97l) provides the key tool for proving the weak convergence of the weighted empirical process. In this appendix we present some notation, terminology, and results that can be found in Bickel and Wichura (1971); although we occasionally make statements in a slightly more general form than Bickel and Wichura. To begin with, let k denote a positive integer and T = T1 x T2 X...x Tk where for each 5 = l,2,...,k, Tj is either a finite subset of (-w,m) or a closed bounded interval in (-m,m). Furthermore, let (4.l) aj = 1nf Ti and bi = sup Tj for J = l,2,...,k. The lower boundary of T is defined to be the set (4.2) R(T) = {t = (t],...,tk) e T : tj a‘j for some j = l,2,...,k}. A block in T is a set B of the form (4.3) B = (s,t] n T where s,t e T, s 5_t, and (s,tJ = (5],t13 X...x (sk,tk]. We say two disjoint blocks A = (s,tJ n T and B = (u,v] n T are neighbors if s agrees with th u and t agrees with v except in the j coordinate (for some ' = l,2,...,k where eithe . . = . . . . = . .. J ) rngtJ u‘15vJ or ujng sJ_<_tJ 34 -- 35 We next introduce a stochastic process X(t), t E T whose For state space F is a normed linear space having norm (.1’) j = l,2,...,k and t e Tj we define the stochastic process Xt having Parameter set T(J) = T1 X...x TJ._1 x Tj+1 X...x Tk by (.i) = (4.4) Xt (s) X(s],...,sj_],t,sj+],...,sk) - (3') Where 5 - (S‘I’ooo’Sj_]’5j+-lgooogsk) E T o For j = l,2,...,k and s,t,u E Tj with s 5_t 5_u we define (4.5) mj(s,t,u)(X) = min{uxgj) - x£5)lL, HX£j) - xéj)nm} and (4.6) M3(X) = sup{mj(s,t,u)(X) : s,t,u e Tj and s 5_t 5_u}. Finally, we define the increment of X around the block B = (s,tJ n T by k-(a +...+5 ) (4.7) X(B)= X (4) ‘ " k X(s + 6(t-s)) 6€{0.1} where 6 = (61,62,...,6k) and s + 6(t-s) = (51 + 61(t] - 51),...,sk + 6k(tk - Sk))' We now state two results from Bickel and Wichura (1971) which will be used to prove Theorem (3.2.1). Lemma 4.l. If X(t) 0 for t 6 R(T), then IA |X(b)| + k max M3(X) lipsk (4 8) sup |X(t)l teT where b = (b],...,bk) and bi = sup Tj, j = l,...,k. Egggf. See (l) on page 1657 in Bickel and Wichura (197l). U 36 Theorem 4.1 (see Theorem 1, page 1658 in Bickel and wichura (1971)). Assume X(t) = 0 for t e £(T) and (4.9) P(minilxmu. wam z A) 5. 118(1) 0 MN for some numbers v > 0, B > 1, and some nonnegative finite measure u on T, and all A > 0 and every pair A,B of disjoint neighbor- ing blocks in T. Then for all A > 0 we have Ck(B.A) (4.10) P(M3(X) _>_ A) _<_——— uB(T \ 2(1)). 3 =1.2.....k AY kck(BsY) (4.11) P( max mgm :1) _<_ 118(T \ mm) isisk where -1 (4.12) CH8“) = 28%) - (%)%;J-U+Y) and (4.13) Ck(B.v) = c,(e.y)ti + (k-1)CL{}+Y(B,Y)J‘*Y. k = 2.3.... Proof. With one minor change (see remark (2) below) Theorem 4.1 follows from the proof of Theorem 1 in Bickel and Wichura (l97l). U Remarks concerning Theorem 4.1: (l) The inequalities in Theorem 1 (Bicke1 and Wichura (1971)) which are analogous to (4.10) and (4.ll) in Theorem 4.1 have u(T) appearing instead of u(T \ £(T)). Furthermore, Bickel and Hichura assume u(£(T)) = 0. However, with one minor change in the definition of F in Step 3 of Bickel and Wichura's proof 37 of Theorem 1, it is seen that the assumption u(t(T)) = 0 is superfluous and also that the inequalities hold with p(T \ £(T)) in place of u(T). This gives a slightly sharper inequality with one less assumption imposed on u. (2) The change referred to in the above remark is to define F in Step 3, page 1660 in Bickel and Wichura as follows: Let F be linear over [tj_],tj], j = l,2,...,m with P(to) = o. F