IN THE NON-I.D. CASE. Pranab Kumar Sen. Department of Biostatistics University of North Carolina at Chapel Hill

Transcription

1 INVARIANCE PRINCIPLES FOR U-STATISTICS AND IN THE NON-I.D. CASE VON MISES' FUNCTIONALS by Praab Kumar Se Departmet of Biostatistics Uiversity of North Carolia at Chapel Hill Istitute of Statistics Mimeo Series No Jue 1982

2 INVARIANCE PRINCIPLES FOR U-STATISTICS AND VON MISES' FUNCTIONALS IN THE NON-I.D. CASE By PRANAB KUMAR SEN Uiversity of North CaroZia, ChapeZ BiZZ For idepedet but o ecessarily idetically distributed radom vectors, wea as well as strog ivariace priciples for V-statistics ad vo Mises' differetiable statistical fuctios are established uder appropriate regularity coditios. Martigale characterizatios of some decompositios of these statistics playa fudametal role i this cotext. 1. Itroductio. Let {X.,i>l} be a sequece of idepedet radom vectors (r.v.) -~ with distributio fuctios (d.f.) {F.,i>l}, all belogig to a commo class 1 - of d.f.'s. Let g(xl,,x m ) be a Borel-measurable fuctio (erez) of degree m (~ 1), ad, without ay loss of geerality, we may assume that g(.) is a symmetric fuctio of its m argumets (vectors). For ~ m, a statistic of the form (1. 1) g(x.,..,x. ) (where C ={l<il< <i < } ) m,m - Iis termed a U-statistic.A closely related oe, termed the vo Mises' differetiabze statisticaz fuctio I is the followig (1.2) -m V = L = Ll =1 g(x.,,x. ) m Whe all the F. are the same, i.e., the X. are idetically distributed (i.d.) r.v., 1 1 cetral limit theorems for these statistics have bee elaboratedly studied by vo AMS Subject Classificatio : 60F17, 62E20 Key Words &Phrases: fuctioal cetral limit theorem, martigale(-differece), Sorohod-Strasse embeddig of Wieer process, tightess. Wor partially supported by the Natioal Heart, Lug ad Blood Istitute, Cotract NIH-NHLBI-7l-2243-L from the Natioal Istitutes of Health.

3 Mises (1947) ad Hoeffdig (1948); asymptotic ormality results i the o-i.d. case were also cosidered by Hoeffdig (1948). Fuctioal cetral limit theorems for these statistics have bee studied by Loyes (1970) ad Miller ad Se (1972), amog others, ad, based o the Sorohod-Strasse embeddig of Wieer processes, Se (1974) has obtaied a strog ivariace priciple for these statistics ; all these results are cofied to the i.d. case where a special decompositio of U, due to Hoeffdig (1961), ad the basic reverse martigale property of U-statistics playa vital role. I the o-i.d. case, either the reverse matigale property of U martigale structure uderlyig the Hoeffdig (1961) or the decompositio may hold, ad the simple proofs of the fuctioal cetral limit theorems i Miller ad Se (1972) or Se (1974) may ot worout well. Nothig particularly has bee doe, so far, o such ivariace priciples i the o-i.d. case (though the case of statioary depedet sequece has received cosiderable attetio durig the past few years). The basic purpose of this study is to icorporate a sequetial sitio which eable us to use some martigale-differece represetatios for these statistics, ad this provide a easy approach to the study of the desired ivariace priciples, uder o extra regularity coditios. Alog with the prelimiary otios, the basic results o U-statistics are stated i Sectio 2 while their derivatios are preseted i Sectio 3. Sectio 4 deals with the parallel results for vo Mises' fuctios. For simplicity of presetatios, throughout the paper, the specific case of m=2 has bee codidered. The case of m=l is trivial, while, for m > 3, a very similar but more legthy treatmet holds. 2. Prelimiary otios ad basic results for U-statistics. For every i,j~l. let (2.1) decompositio of U (viz., Se (1960)) alog with the Hoeffdig (1961) decompo -2-

4 The, by (1.1) ad (2.1), (2.2) -1 e() = EU = (2) ~l~i<j2? e.., for every > 2. 1) -- Our first goal is to study wea ivariace priciples relatig to the (partial) _ ~ sequece { \(U -6());2<<} ad the tail-sequece { (U 6()), ~ }, which would exted the results of Miller ad Se(1972) ad Loyes(1970) to the o-i.d. case. Also, we lie to exted the results i Se (1974) to the o-i.d. case. As i Hoeffdig (1948), we let for every i( f j) ~ 1, (2.3) 1jJl (. ). (x.) = Eg (x., X.) - 6.., 1jJ2 (.. ) (x.,x.) = g (x.,x. ) - e.. 1) 1 1) 1) 1) 1) 1) 1) (2.4) I'; (..).... = E1jJ (..).. (X.,..., X. ). C 1 1 l C lc+l 12;)c+l J2 c l I. 1 C l c + l l C 1jJ (..).. (X., OJ X. ) C 1 1. l C J c + l )2 1 1 l C for all possible combiatios of distict i, j ad c=1,2. For a fixed c(=1,2) r r ad ( ~ m), the average over the possible I'; 's with 1 < i,j < ad distict c - r r-- i j is deoted by I';c The, from Hoeffdig (1948), we obtai that r' r ', (2.5) E{U 6 }2 = ()-l ~2 (c) (-2) > 2. - () 2 c=l 2 2-c I';c, { } { -1 }~ The asymptotic ormality of U - 6() I I';l, uder some extra (mild) regularity coditios, has bee established by Hoeffdig (1948). I particular, the followig coditios relate to the asymptotic ormality result o Let (2.6) Assume that there exists a umber A, such that for every ~ 2, (2.7) sup i,j~l Further, (2.8) J J g2(x l,x 2 )df. (x)df.(x) < A < 00 1 J -- sup max I - /2+0 l<i< E 1jJl(i),(Xi ) < 00 ad (2.9) lim { /- / /2} ~. IE 1jJl (. ) (X. ) I {~. le1jjl (. ) (X.)} = 0, ~ 1= 1, 1 1= 1, 1 where 0 (0 < o~l ) is somepositive umber; Hoeffdig(1948) too 0 = 1.I particular, (2.9) isures that l';l goes to 00, as + 00, though possibly i a arbitrary, -3-

5 maer. I the i.d. case, ~l(i),(x) =~l (x) does ot deped o (i,) ad ~l, ~l is also idepedet of. Hece, we may,istead of (2.8)-(2.9), appeal to the 4It classical Lideberg coditio, ad this will oly require that ~l > O. Note that by (2.5), (2.6) (2.10) 2 E{U The igeuity of the ad (2.9), 2-8()} ={4 ~l HI + o(l)}, -2 }{ } = 4{2::, 1 E1/Jl(') (X,) 1 + 0(1) 1= 1, 1 Hoeffdig (1948) approach lies i the (quadratic mea) - approximatio of [U - 8( )] by 2 L, 1 1/Jl(') (X,) ad icorporatig the cetral 1= 1, 1 limit theorem for the triagular array {y, = ~l(') (X.), i=l,,; ~ I}. 1 l., 1 For our desired ivariace priciples, we eed to establish some maximal iequalities relatig to { [U - 8()]-2 L~=lYi ;~ 2} ad some ivariace priciples relatig to the triagular array { S = L, ly., < ; > 2}. Towards this, 1= l. we have the followig. Theorem 2.1. (I) If lim -t<x> {(log )/C~l )} = 0, the, for every s >0, uder (2.7),, { max I I~} (2.11) P ~ (U - 8())- 2S > se ~l,) + 0, as + 00, lim { - 1 L -I} _ while, if -t<x> =l (~l ) - 0, the, as, + 00, (2.12) max -~ I I P{ ~ ~l, (U - 8()) -2S > s 2 } + O. (II) If (2.13) while, if lim {(~ )-1}= 0, the, for every s > 0, as + 00, -t<x> 1, p{ sup I(U > - 8 ()) ~:: { L>(3~1,)-1 } -1 se ~l )2, o, the, as } + 00, p{ sup -1 (2.14) ~~~l (U - 8 ()) - 2 S I > s 2 } + >, O. (III) If L>2 (2~ f1 < 00,the, for every S > 0, as + 00,,l - (2.15) p{ sup I{(U _ 8 ()) -2\} I/ (~l ) 2 > s } + > O., The proof of the theorem is cosidered i Sectio 3. The maximal iequalities i (2.11) - (2.15) alog with proper costructio of stochastic processes lead us { } -1 to the desired results. I the i.d. case, S; > 1 (or { S; > 1 } ) 0, -4-

6 form a martigale (or reverse martigale) sequece, ad hece, the forward ( or bacward) ivariace priciple for U-statistics, cosidered by Miller ad Se (1972) ad Loyes (1970), follows directly by a appeal to the fuctioal cetral limit theorem for martigales (or reverse martigales). I the o-i.d. case, either the forward or the reverse martigale property holds, ad hece, a more elaborate treatmet is ecessary. Note that i order to establish wea ivariace priciples, oe eeds to study the covergece of fiite dimesioal distributios (f.d.d.) ad the tightess of appropriate stochastic processes costructed from these statistics. By virtue of Theorem 2.1, it suffices to costruct such processes -1 from the S or S' I this respect, if we cosider arbitrary c(~ 1) ad O<t l < <t «1), the, for ay o-ull c- [it.]), j=l,,c, Z J of the defiitios of the A = (Al'".,A c )' o lettig j = [t j ] ( or ~c, ~c, -1. I.J. 11\.S (or 1.J._ll\.. S ), we may, by virtue J= J j j J- J J j j S ad Y j, express Z as (Zl+'''+Z) (or Zll+'" + Z ) where the Z. form a triagular array of row-wise idepedet r.v. 's. l l l Thus, the Liapouoff-type coditio i (2.8)-(2.9) may agai be called o to verify the asymptotic ormality of Z Thus, the covergece of f.d.d. 's may be established uder coditios similar to (2.8)-(2.9). However, these fiite dimesioal distributios, though asymptotically multiormal, may ot coform to that of a Wieer process, particularly whe ~l, does ot coverge to a positive costat. Thus, i the o-i.d. case, for the ivariace priciples to be studied, the questio of havig a Wieer process i the picture remais of some good iterest. Secodly, eve if, such a Wieer process does ot come to the picture, the questio of wea covergece to some appropriate Gaussia fuctio eeds to.. be addressed. I both the cases, the basic tas is to establish the tightess property of the stochastic processes uder cosideratio, ad, the to examie the covariace structure to decide whether the Gaussia fuctio is of Wieer structure or ot. This will be studied here. Let us itroduce the followig stochastic processes z~j) = {Z~j)(t);O<t~l}, -5-

7 j=1,...,4, all defied o the space DI 0, lj, where (2.16) Z(1) (t) = {O, 0~t<2/. e([t]))/2{s l, p. [t] (U [t], t2:. 2/ ; (2.17) Z(2) (t) ~ (1) = {sl,isi, [t]} Z (t), 2/~t~l, ad 0, otherwise; ~ ~ (2.18) Z(3) (t) = 2{U mi{ :/~ t}, (t) e((t))}/2s{, et) = O~t~l ; (2.19) z(4)(t) ~ (3) {Sl,/Sl,(t)} Z (t) O<t<l, Our goal is to study wea coveegece of these processes to appropriate Gaussia fuctios. (2.20) (2.21) (2.22) Correspodig to the d.f. {F.,i> I} ad the erel g(x,y), we defie F( ) = L. IF., for > 1, 1= 1 - gle) (x) = J g(x,y)df() (y), Sl (F()) = J g~()(x)df()(x) - (JJ g(x,y)df() (x)df() (y) )2 ad (2.23) 2 ~ -1 = Li=l( - - J gl()(x)d[f i (X)-F() (x)] 2 ) The, followig Hoeffdig (1948) ad Se(1969), we have 2-1 (2.24) ~2 sl = sl(f()) - ~ + o( );, ~ sl(f()) As such, if F() coverges to some F, sl (F) is a cotiuous fuctioal of F i some eighbourhood of F ad ~2 coverges to some ~2 < sl (F), the, sl, -+ S T (-F) _ A 2 > a ~l D, as. There may be other situatios where Sl may coverge, to some positive limit S without requirig that F() coverges to a limit F. Theorem 2.2. If (2.7)-(2.9) hold ad sl -+ s > 0, the, for each j(=1,,4),, (.) Z J coverges i law to Z = {Z(t);O<t< I}, a stadard Wieer process o [0,1]. -- The result holds if i (2.16)-(2.19), 8() is replaced by 8(F()) =JJ g(x,y)df() (x). df()(y), for all possible. I the other cases where sl may ot coverge to a positive limit, though the, f.d.d. 's of the Z(j) may coverge to some multiormal distributios, the covariace fuctio may ot coform to that of a Browia motio. Nevertheless, we may lie to study the tightess property of these processes o I the sequel, it will be assumed~ -6-

8 that ~l is t i. Also, let h (i) = ~l I~l ' <. Note that h (i),,, > i, 'd ~. We may also eed the followig (2.25) (2.26) for some 0: max{ h (i) : < } - Ih (q/) - h (i) I <Klq/ - - = 0(1), /lo: > llr, for every, ad q, such that q/> i ~ 8 >0, where K may deped o 8 but ot o. Theorem 2.3. If Um ~l --, >o~ the~ uder (2.7)-(2.9), the tightess of z(l) ad z(3) are i order~ whize~ if i additio~ (2.25) ad (2.26) hozd~ the Z(4) are azso tight. Whe lim ~l, z(2) ad may ot be positive, we may still be able to establish the tightess property of the z(j), provided we impose more striget coditios o the erels. For every ~ q (2.27) Z ;q, > l~ 2, we defie -~{ - = L:. 1 [1jJl (. ) (X. ) 1= 1,q 1 Theorem 2.4. If for some r > 2~ for every,q ad ~ we have (2.28) Elz I r / {EZ 2 }r/2 < K < the~ ;q~ ;q~ (.) the tightess property of the Z J without the coditio that Zim ~l > O. --, hozds uder the hypothesis of Theorem 2.3~ Fially, we cosider the strog ivariace priciple for V-statistics. I Se (1974), the basic tool of martigales ad reverse martigales were icorporated i the derivatio of the mai result. I the o-i.d. case, these martigale or reverse martigale property may ot hold ad hece that method may ot be applicableo By virtue of (2.15), the desired strog ivariace priciple would follow if we ca establish the Sorohod-Strasse embeddig for the sequece {S; ~ ~ 2}. However, for a double sequece of idepedet (row-wise) r.v., we may ot have such a powerful result to utilize i the curret cotext. Keepig i mid (2.6) ad (2.21), we coceive of a sequece{1jj(x ) ; 2:.l} of radom variables ad assume that there exists a positive c > 1, such that (2.29) = L:~ 1 E [ ~l (. ) (X. ) 1= 1, 1-7-

9 2 ad that s = I=l as 7 00 More specifically, we assume that (2.30) lim 700 1'; > 0 Typically (2.29) ad (2.30) hold whe F()!g(x,y)dF(y) -!!g(x,y)df(x)df(y), ~ 1, 7 F as 7 00, so that ljj (X) where the 1JJ (X ) eed ot be i.i.d. (as i the multisample situatio where the F i belog to a fiite class {Gl,,G m } of d.f. 's.). Let Z {Z(t):O 2 t < oo} be a radom process o [0,00 ), where Z(t) = Z() for < t < +l ad (2.31) Z () = 0, < 2, 1 = (U - 8(F()))/2(1';)~, ~ 2. Fially, let W = {W(t):O 2 t < 00 } be a stadard Wieer process o [0,00 ). Theorem 2.5. Uder (2.29) ad the hypothesis of Theorem 2.2~ (2.32) : Z(t) = Wet) + oct 2 ) almost surely~ as t 7 00 The result ca be exteded to the case where s may ot go to 00 at the rate : of 2, but at a slower rate. I that case, we eed stroger order of covergece i (2.29). We will ot elaborate it further. 3. Proofs of the theorems. We start with the followig (sequetial) decompositio of U (3.1) [ c.. Se(l960)]. For m=2, by (1,1), (2.1), (2.2) {i-l } (2)[ U - 8()] = I i =2 Ij=l [g(xi,x j ) - 8 ij ] = I i =2 ad (2.3), i-i I. 1 J= 1JJ 2 (.. ) (X., X. ) 1J 1 J Further, we write, for every i ~ 2, (3.2) ~i-l i-i i-l{ U i = '"'j=l ljj l (i)j (Xi) + Ij=l ljj l (j)i (X j ) + Ij=l 1JJ 2 (ij) (Xi,X j )- ljj l (i)j (Xi) = U il + U i2 + U i ' say. - ljj l (j)i (X j )} Let ~ be the sigma-field geerated by (3.3) ad E (U il however, (Xl,,X ), for ~ &>. 1) = 0, \J i > 2 ; The, ote that this martigale (differece) property does ot hold for the U Also, i2 ote that for every > 2, -8-

10 (3.4) -1 - (-l) L. 2{Uo l +U 2} = L 1 '/'1(0) (Xo) = L ly = S, 1.= = 'Y 1., - 1. = where the Y ad S are defied after (2.10). Hece,for every ~ 2, - -1 (3.5) [U - S()] - 2S = 2(-l) L i =2 U i = R, say, ad, covetioally, we let R l = O. Note that by (3.3) ad (3.5), (3.6) E[IR I IcB _ l ].:.IE(R I Q)-l) I = ~=i IR_ll, \j.:. 2. Therefore, by the Birbaum-Marshall (1961) iequality, for every sequece {a } of positive umbers, t > 0 ad l' 2 : 2 > l ~ 2, { max I I } (3.7) P << a R >t < t L= (b - (-1) b + 1 )ER ' where { -I -I} (3.8) b = max a, (-1)a + 1,,( 2 -l) (-l)a 2 Note that by (3.20, (3.3) ad (2.7),for every ~ 2, (3.9) ER = 4(-l) L =2E{(U )}.5.8A (-l) L i =2(i-l) = 4A/(-1).5. 8A. ;,,; Thus, if we let l =2, 2 =, a=1,2~~ (so that b=l, ) ad t= E(~l ) 2,, the, (2.11) follows from (3.7) ad (3.9) whe (~l )-llog + 0 as + 00., ;,,; -~ For (2.12), we tae t= E 2 ad a = ~l, ad the proof follows o parallel lies. -1;"; 1 For the proof of (2.13), we tae 1 =, , t = E( ~l,) 2 ad a = -, for >. Note that (~l,) L > ER = O(~l, L> ) = O((~l,) ), ad hece, (3.7) isures(2.13) uder the assumed coditio. The proofs of (2.14) ad (2.15) are very similar, ad hece, omitted. This completes the proof of Theirem 2.1. I the proof of Theorem 2.2, we may ote first that by (2.1), (2.2), (2.7) ad (2.20), (3.10) 18() - 8(F())I.5. (_l)-la~, for every.:. 2, so that max{ I8()- 8(F())I// :2<<} ad max{ ~ 18()- 8(F())I :~ } both coverge to 0 as + 00 Hece, we may use 8() ad 8(F()) iterchageably Sice the covergece of the f.d.d. 's follow alog the lie of Hoeffdig (1948, Theorem 8.1), we shall oly prove the tightess property of the z~j), j=1,,4. For this purpose, we defie for each.:. 2, -9-

11 (3.11) (3.12). -1 i W = 1 L. 2(1-1) L. 1 lji1(i)j(\) = L i =2 lji1 (i-i) (\), say 1= J=. -1 i W = 2 Li =2 (1-1) L j =1 lji1(j)i (X j ) = L. 2, iij1(i-1)i say. 1= The, by (3.1), (3.2), (3.11) ad (3.12), for every q ~ ~ 2, we have (3.15) -1 q -1 (q-1) L i =2 U ij - (-1) L i =2 U ij -1 q -1 (3.13) = (W qj - W j ) - (q-1) Li=2Wij + (-1) Li=2Wij, for j =1,2. By (3.4) ad (3.13), for every q ~ ~ 2 ad, -~I I 2 -~! I I 1 q 1 I (3.14) (S - S) 2. L. I {W. - W J ' +~q_1l1'=2 W. '--1L. 2W" }, qq J = qj 1J - 1= 1J Now, uder (2.8), for some 0 > 0, Similarly, (3.16) where, -(1+0/2)E! W _ W = -(1+0/2)E!L q iij. (X.) q (1-1) 1 < (_)1+0/2 -(1+0/2) {( _)-l L q E I;J; (X ) 12+0 } c 2+0 q q +1 ~1(i-1) i < C [(q_)/]1+0/2, where c ad C are fiite positive costats. -(1+0/2)E! 12+0 _ -(1+0/2) 1 q W q2 - W 2 - E Li =+1 lji1(i-1)i < - (1+0/2) (L~ i 1+0/2 E liij ) (L~ i - (2+0) /2 (1+0)) 1+0-1=+1 1(1-1)1 l=+l, /2 uder (2.8), E ljil(i-l)i = 0(1 ), so that by some stadard steps, the right had side of (3.16) ca be bouded by 1+0/2 (3.17) C[(q-)/], for every 2. q 2. ; C < 00 Also, ote that (3.18) -1 q -1 (q-1) L i =2 W ij - (-l) L i =2 W ij = [(q-)/(q-l)]{(q_)-ll~ lw" - (_l)-ll~ 2 W.. }, for j=1,2. 1= + 1J 1= 1J Thus, uder (2.8), a momet-boud similar to (3.15) ad (3.17) applies to (3.18) as well. Usig (3.4) ad (3.13) through (3.18), we coclude that uder (2.8), for every > q > > 2, we have (3.19) El-~ (S - S ) < qq K[(q_)/J l + 0 / 2, 0 > 0, where K is a positive costat, idepedet of. By virtue of Theorem 12.3 of Billigsley (1968) ad our Theorem 2.1, (3.19) isures the tightess of Z(l) whe -10-

12 ~l + ~ > O. The tightess of Z(2) follows from (2.17) ad the tightess of, Z~l) where we use the fact that for every t E (0,1], ~l,/~l,[t] + 1, as For the process Z(3) i (2.18), we agai use Theorem 2.1 ad ote that ~ -1-1 ~ (3.20) ( S -q Sqq) = [( -q )S - q (Sqq-S)]' As such, proceedig as i (3.14) through (3.19), we obtai that for every < < q < 00, uder (2.8), for some 0 > 0, < K[(/_/q)]1+O/2, where K«00) does ot deped o. Thus, Theorem 12.3 of Billigsley (1968) may agai be recalled to verify the tightess of Z(3) The tightess of Z(4) follows from (2.19), the tightess of Z~3) ad the fact that ~l, + ~ > 0, as + 00 This completes the proof of Theorem 2.2. To prove Theorem 2.3, we ote that Theorem 2.1, (3.19) ad the assumptio that Ijm ~l, > 0 isure the tightess of Z~l), while (3.21) ad the other coditios isure the tightess of Z(3) Further, (2.25)-(2.26) ad the tightess of Z(l) - ( or Z(3) ) isure the tightess of Z(2) ( or Z(4) ). Hece, the details are omitted. (3.22) For Theorem 2.4, we may ote that uder (2.28), for some r > 2, for every 0< s ~ t ~ 1, where K «00) does ot deped o. A similar mometboud holds for the Z(3) also. Hece, isure the tightess of Z(l) ad Z(3) Theorem 12.3 of Billigsley (1968) ad (3.22) The case of Z(2) ad Z(4) follows by usig (2.25)-(2.26) alog with the tightess of Z~l) ad Z~3) Fially, we cosider the proof of Theorem 2.5. Note that by (2.15) ad Theorem 2.2, for every positive E, as + 00, (3.23) O. Let us defie ow (3.24) = l:. 1 1= 1jJ. (X.), _> Note that the S ivolve idepedet (but, possibly o-i.d.) summads, ad, by -11-

13 (2.29), (3.24) ad the defiitio of the S ' we have 1 1 {sup ->2->2 IS - S > s } P 2. t:l, (3.25) 1 ~ I> P{(t: l )-~IS - S > s } -, < I -1 (t: )2 - > l, ) E( S - S I -1 I -1 [O(-l(log )c) ] > (t:l,) S > t: l, where c > 1 ad the t: l coverge to a positive limit t: Hece, the right had, side of (3.25) coverges to 0 as -+ co. Thus, from (3.23) ad (3.25), we obtai, that as -+ co (3.26) 1 (t: l )->2 I [U, O the other had, for the - e(f())] -2S: I -+ sequece {S~ ;.:::.. l} o, almost surely., we directly appeal to the Sorohod-Strasse embeddig of Wieer process [ c.f. Strasse (1967)] ad coclude that uder (2.8), (3.27) S /ft, ~ We) + o( 2 ) almost surely, as -+ co. Thus, (2.31), (3.26) ad (3.27) imply (2.32). Q.E.D. 4. Ivariace priciples for the vo Mises' fuctioal. Recall that by (1.1) ad (1.2); (4.1 ) I additio to (2.7), we ow assume that (4.2) sup i>l Let us ow write 2! g (x,x)df. (x) 1. < B < co (4.3) (4.4) e~ = Eg(X.,X.) ad t:~ = eo = -ll~ e~ ad 1=1 1 Var{g(X.,X.)}, i > e -;:;-OS - S 2 () = ()'.:::.. The, uder (4 0 2), by the Kolmogorov law of large umbers, as -+ co (4.5) -+ 0, almost surely Further, as i (3.21), for some 6 >0, (4.6) E!-lSI2+6 = O(- 1-6 / 2 ), for every.::: Hece, by (2.15), (4.6) ad the Borel-Catelli Lemma, uder (2.8), (4.7) U Se) -+ 0 almost surely, as -+ co -12-

14 Therefore, by (4.1), (4.5) ad (4.7), we obtai that as (4.8) (U - V ) - eo + e, ) I -+ 0, almost surely. -+ oo, -1 -:ct) Cosequetly, o lettig e() = [(-l)e() + e ] ad otig that (V - e()) = (U - e()) + {(V - U) - ~ + e() }, we coclude that as -+ 00, (4.9) ( V - e() ) - ( U - e()) -+ 0 almost surely. Thus, if i (2.18),(2.19) ad (2.31), we replace the (U - e()) by the (V e ) d d hi by zo(3). Zo(4) ad ZO. - () a eote t e resu tlg processes ' ' respectively, the the ivariace priciples i Theorems 2.2, 2.3, 2.4 ad 2.5 hold for these processes as well. Similarly, uder (4.2), { -~ I -1 I } P (4.10) max ~i=lg(xi,xi) - e() : , as -+ 00, while, by the Theorems i Sectio 2, 1 (4.11) max{ -~ IU - e() I : < } p -+ o, as Cosequetly, uder (4.2) ad (2.7)-(2.9), as (4.12) 1 P max{ -~ I (V - e()) - (U - e()) I : < } -+ O. Thus, if i (2.16)-(2.17), we replace the U - e() by V - e(), ad deote the resultig processes by Zo(l) ad ZO(2), respectively, the, uder (4.2) ad the hypotheses of Theorems 2.2, 2.3 ad 2.4, the ivariace priciple holds for ZO(l) ad Zo(2) also. This leads us to the followig. Theorem 4.l.Uder the additioal assumptio (4.2)J Sectio 2 hold for the vo Mises' fuctioals as well. REFERENCES the ivariace priciples i.. BILLINGSLEY, p. (1968). Covergece of Probability Measures. New Yor: Wiley. BIRNBAUM, Z.W. ad MARSHALL, A.W. (1961). Some multivariate Chebyshev iequalities with extesios to cotiuous parameter processes.a.math.statist. 32 J HOEFFDING, W. (1948). A class of statistics with asymptotically ormal distributio. A.Math.Statist. 19 J HOEFFDING, W. (1961). The strog law of large umbers for U-statistics. Ist. Statist Uiv. North Carolia J Mimeo. Report. No LOYNES, R.M. (1970). A ivariace priciple for reverse martigales. Proc.Amer. Math. Soc. 25 J

15 MILLER, R.G.Jr., ad SEN, PoK.(1972). Wea covergece of U-statistics ad vo Mises' differetiable statistical fuctios o A.Math.Statist.43, SEN, P.K. (1960). O some covergece properties of U-statistics. Calcutta Statist.Assoc.Bull. lo,1-19. SEN, P.K. (1969). O a robustess property of a class of oparametric tests based o U-statisticsoCalcutta Statist.Assoc.Bull.l8, SEN, P.K. (1974). Almost sure behaviour of U-statistics ad vo Mises' differetiable statistical fuctios.a.statist. 2, SEN, P.K. (1981). Sequetial Noparametrics: Ivariace Priciples ad Statistical Iferece. New Yor: Wileyo STRASSEN, V. (1967). Almost sure behaviour of sums of idepedet radom variables ad martigales. Proc. Fifth Bereley Symp. Math.Statist.Probability. 2, ( ed: L.LeCam et o ai, Uiv. Calif. Press). VON MISES, R. (1947). O the asymptotic distributio of differetiable statistical fuctios. A.Math.Statist. l8,