EXPONENTIAL AND MOMENT INEQUALITIES FOR U-STATISTICS

Transcription

1 EXPONENTIAL AND MOMENT INEQUALITIES FOR U-STATISTICS EVARIST GINÉ, RAFA L LATA LA AND JOEL ZINN ABSTRACT A Bernsten-tye exonental nequalty for (generalzed) canoncal U-statstcs of order s obtaned and the Rosenthal and Hoffmann-Jørgensen nequaltes for sums of ndeendent random varables are extended to (generalzed) U-statstcs of any order whose kernels are ether nonnegatve or canoncal 1 Introducton Exonental nequaltes, such as Bernsten s and Prohorov s, and moment nequaltes, such as Rosenthal s and Hoffmann-Jørgensen s, are among the most basc tools for the analyss of sums of ndeendent random varables Our obect here conssts n develong analogues of such nequaltes for generalzed U-statstcs, n artcular, for U-statstcs and for multlnear forms n ndeendent random varables Hoffmann-Jørgensen tye moment nequaltes for canoncal (that s, comletely degenerate) U-statstcs of any order m were frst consdered by Gné and Znn (199), and ther verson for U-statstcs wth nonnegatve kernels turned out to be useful for obtanng best ossble necessary ntegrablty condtons n lmt theorems for U-statstcs (By Khnchn s nequalty t s rrelevant whether one consders canoncal or nonnegatve kernels n moment nequaltes, at least f multlcatve constants are not at ssue) Klass and Nowck (1997) also obtaned moment nequaltes for nonnegatve generalzed U-statstcs, but only for order m =, and ther decomoston of the moments s more comlete than that n Gné and Znn (199) Ibragmov and Sharakhmetov (1998, 1999) recently obtaned analogues of Rosenthal s nequalty for nonnegatve and for canoncal U- statstcs The moment nequaltes we resent n the frst art of ths artcle, vald for canoncal and for nonnegatve generalzed U-statstcs of any order m, when secalzed to m =, reresent the same level of moment decomoston as the Klass-Nowck nequaltes, concde wth thers for Research artally suorted by NSF Grant No DMS Research artally suorted by Polsh Grant KBN PO3A

2 Inequaltes for U-statstcs owers > 1 (excet for constants) and are exressed n terms of dfferent, smler quanttes for owers < 1 Prooston 1 below, whch consttutes the frst ste towards more elaborate bounds such as those n Theorem 3 below, has also been obtaned, u to constants, by Ibragmov and Sharakhmetov Our roofs consst of smle teratons of the classcal moment nequaltes for sums of ndeendent random varables The moment nequaltes n the frst art of ths artcle do mly exonental bounds for canoncal U-statstcs of any order and wth bounded kernels whch are sharer than those n Arcones and Gné (1993); however, they are not of the best knd as they do not exhbt Gaussan behavor for art of the tal, whch they should n vew of the tal behavor of Gaussan chaos In the second art of ths artcle we mrove the moment nequaltes from the frst art n the case of generalzed canoncal U-statstcs of order, and for moments of order (Theorem 3) The bounds not only nvolve moments but also the L oerator norm of the matrx of kernels Then we show how these mroved moment nequaltes mly what we beleve s the correct analogue (u to constants) of Bernsten s exonental nequalty for generalzed canoncal U-statstcs of order (Theorem 33) Ths exonental nequalty, whch does exhbt Gaussan behavor for small values of t, s strong enough to mly the law of the terated logarthm for canoncal U-statstcs under condtons whch are also necessary The man new ngredent n ths art of the aer s Talagrand s (1996) exonental bound for emrcal rocesses, whch gves a Rosenthal-Pnels tye nequalty for moments of emrcal rocesses (Prooston 31) basc for the dervaton of the moment nequalty for U-statstcs of order Because of the decoulng results of de la Peña and Montgomery-Smth (1995), we can work wth decouled U-statstcs, and ths allows us to roceed by condtonng and teraton Moment nequaltes We consder estmaton of moments of generalzed decouled U-statstcs, defned as (1) h 1,, (X (1) 1,, X (m) ), 1 1,, n where the random varables X () : 1 n, 1 m, m n, are ndeendent (not necessarly wth the same dstrbuton) and take values n a measurable sace (S, S), and h 1,, are real valued measurable functons on S m For short, ths sum s denoted by h Gven J {1,, m} (J = s not excluded), and = ( 1,, ) {1,, n} m we set to be the ont of {1,, n} J obtaned from by

3 Gné, Lata la and Znn 3 deletng the coordnates n the laces not n J (eg, f = (3, 4,, 1) then {1,3} = (3, )) Also, ndcates sum over 1 n, J (for nstance, f m = 4 and J = {1, 3}, then h = {1,3} h 1,, 3, 4 = 1 1, 3 n h 1,, 3, 4 (X (1) 1,, X (4) 4 )) By conventon, a = a Lkewse, whle E wll denote exected value wth resect to all the varables, E J wll denote exected value only wth resect to the varables X () wth J and {1,, n} By conventon, E a = a Rosenthal s nequalty s easest to extend to U-statstcs because t nvolves only moments of sums (as oosed to moments of maxma and quantles for Hoffmann-Jørgensen s nequalty) So, we wll frst obtan analogues of Rosenthal s nequalty, and then we wll transform these nequaltes nto analogues of Hoffmann-Jørgensen s by frst showng that some moments of sums can be relaced by moments of maxma, and then, that the lowest moment can n fact be relaced by a quantle We wll llustrate ths threestes rocedure frst n the case of nonnegatve kernels and moments of order 1 Then we wll see that ths also solves, va Khnchn s nequalty, the case of canoncal kernels and moments of order Fnally, we wll consder the case of moments of order < 1 for ostve kernels and < for canoncal, cases n whch the nequaltes are less neat, but stll useful We wll ay some attenton to the behavor of the constants as n these nequaltes snce such behavor translates nto (exonental) ntegrablty roertes 1 Nonnegatve kernels, moments of order 1 For nonnegatve ndeendent random varables ξ, we have the followng two mrovements of Rosenthal s nequaltes, vald for 1: 1) Lata la s, 1997: ) e (R 1 ) E( ξ (e) max Eξ, e( ) ] Eξ, > 1, (see Pnels (1994) for the corresondng nequalty when the random varables are centered); ) Johnson, Schechtman and Znn s, 1985: ( ) (R ) E ξ K ( log ) max Eξ, ( Eξ ) ], > 1, where K s a unversal constant See Utev (1985) and Fgel, Htczenko, Johnson, Schechtman and Znn (1997) for more recse nequaltes of the same tye And for general > 0, we have the followng mroved Hoffmann-Jørgensen nequalty, that follows from Kwaeń and Woyczyńsk (199) and whch

4 4 Inequaltes for U-statstcs can be obtaned as n the roof of Theorem 13 n de la Peña and Gné (1999): 3) (H) E ξ ( 1) 0 ( + 1) +1 t 0 + E max ξ ], > 0, where { } t 0 := nf t > 0 : Pr ξ > t 1 ], and where we wrte norm for absolute value n order to nclude not only ndeendent nonnnegatve real random varables, but also ndeendent nonnegatve random functons ξ takng values n certan rearrangement nvarant saces such as L s (Ω, Σ, µ), 0 < s <, wth ξ := ( ξ s dµ ) 1/(s 1), or l (L s ) Note that, by Markov, t 0 (E 1/r ) r ξ 1/r, so that, (H) becomes: 4) for 0 < r < <, E ( ξ ( 1) 0 ( + 1) +1 /r E ) /r ξ r ] (H r ) + E max ξ Inequaltes (H) and (H r ) hold for saces of functons whch are quasnormed measurable lnear saces whose quasnorm has the roerty that x y whenever 0 x y In the followng rooston we extend nequaltes (R 1 ) and (R ) by means of an easy nducton Prooston 1 Let m N, > 1, and, for all {1,, n} m, let h be a nonnegatve functon of m varables whose -th ower s ntegrable for the law of X = (X (1) 1,, X (m) ) Then, ( ) ] ( ) E J ch E h () max (e ) m E J c ( ) J E J E J ch ], and also, there exsts a unversal constant K < such that ( ) ( ) m ( ) E h K m ( ) max E J E J ch ] log Proof The roof of ( ) wth sum over the subsets J nstead of maxmum dffers from that of () only n the startng ont ((R ) nstead of (R 1 )); c c

5 Gné, Lata la and Znn 5 then, relacng sum by maxmum smly ncreases the constant by a factor of m The left sde nequalty n () follows by Hölder snce 1 Consder the rght hand sde nequalty For m = 1 ths s ust nequalty (R 1 ) and we can roceed by nducton Suose the result holds for m 1 By alyng the nducton hyothess to ( ) ( ) ] E h = Em E {1,,}, {1,,} we only have to ( consder the generc term n the decomoston () for the new kernels h m ) wth the X (m) varables fxed In other words, lettng J be any subset of {1,, m 1} and J c ts comlement wth resect to {1,, m 1}, we must estmate ( ( ) ) E m E J = E J E m c E J c h ( (E J c c h h ) ) Rosenthal s nequalty (R 1 ) aled to the kernels E J c c h wth the varables n J fxed, gves E m ( ( E J c c h ) ) (e ) (, c E m E J c h + E m ( E J c ) c h ) ] Uon ntegratng each term wth resect to E J and summng over, we then obtan ( ( ) ) E m E J (e ) + c E J c E J {m} ( h c {m} E J {m} E J c {m}h ) (E J c c h ) ] Multlyng by (e ) () J, ths s the sum of two terms of the form (e) m J E J ( c E J ch ) (for J = J and for J = J {m}), rovng the rooston

6 6 Inequaltes for U-statstcs Ths rooston solves the roblem of estmatng, u to constants, the moments of a decouled U-statstc by comutable exressons For nstance, f the functons h are all equal and f the varables X () are d, then the tycal term at the rght of (1) ust becomes n J + J c E J (E J ch), a mxed moment of h For m = the rght hand sde of nequalty () s ust: ( ) ( ) E h, (X (1), X () ) (e ) Eh, (X (1), X () ), + +, E 1 ( E ( ) E h, (X (1), X () ) ) ) E 1 h, (X (1), X () ( ) +, Eh, (X(1), X () ) We have been careful wth the deendence on of the constants because t s of some nterest to obtan constants of the best order as In fact, ( ) exhbts constants of the best order as can be seen by takng the roduct of two ndeendent coes of the examle n Johnson, Schechtman and Znn (1985), Prooston 9 Next we relace the external sums of exected values at the rght sde of the above nequaltes by exectatons of maxma wthout sgnfcantly alterng the order of the multlcatve constants If ξ are ndeendent nonnegatve random varables, then, (3) 1 δ 0 ] Eξ I ξ>δ0 where (4) ] E max ξ δ 0 + Eξ I ξ >δ 0, 0 < <, δ 0 = nf t > 0 : Pr { ξ > t } ] 1 (Gné and Znn (1983); see also de la Peña and Gné (1999), age ) The left hand sde of (3) gves that, for 0 < r < and ξ ndeendent, E ξ E max ξ + ( E ξ r)( E max ξ ) ( r)/ (5) (eg, de la Peña and Gné (1999), age 48) Ths nequalty, aled wth r = 1 <, yelds ( ] (6) α E ξ (1 + α ) max α E max ξ, E ξ ) for all α 0 There are smlar nequaltes for other values of r; r = 1 s adequate for ξ 0, but r = s better for centered varables If we use

7 Gné, Lata la and Znn 7 nequalty (6) n ( ), teratvely for the last term, we obtan that, for a unversal constant K (easy but cumbersome to comute), h, 0, > 1, ( ) ( ) ( ) E h, K (e ) 4 Eh, + E 1 max E h, (7),, + E max ( ) ] E 1 h, + E max h,, Inequalty (7) was obtaned, u to constants, by Klass and Nowck (1997) (t s ther nequalty (414)) Our roof s dfferent, and t s contaned n the roof of the next corollary, whch extends nequalty (7) to any m Corollary Under the same hyotheses as n Prooston 1, there exst unversal constants K m such that ( ) ] ( ) E J max E J ch E h (8) max K m c ( ) J E J max E J ch ], and ( ) ( ) m (8 ) E h K ( ) m max E J max E J ch ] log Proof The left sde of (8) follows by Hölder Inequalty (8 ) has a roof smlar to that of the rght hand sde of (8), and therefore we only rove the latter We wll rove t by nducton over m smultaneously wth the nequalty m Eh K ( ) (9) m J E J max E J ch ] c J {1,,m} Let us frst note that the nequaltes (9) for 1,, m 1 together wth () mly (8) It s therefore enough to show that f (8) and (9) hold for 1,, m 1 then (9) s satsfed for m We wll follow the notaton of the roof of Prooston 1 Inequalty (9) for m = 1 s ust (6), and (8) for m = 1 s (H 1 ) (whch also follows from (R 1 ) and (6)) By the nducton assumtons we have m Eh = (10) E m () E {1,,} h {1,,} K J {1,,} ( ) ] ( J +1) E J E m max E J c 1 h J c c c

8 8 Inequaltes for U-statstcs Now, by (6), for any J {1,, m 1} we have (11) ( ( J +1) E J E m max E J c 1 h J c (1 + ) ( J +1) E J {m} max + J E J ( ) {m} E m max ( To estmate the last term we note that ( J E J E m max E J c 1 h J c (1) ( ) J E J E J c {m}h ) E J c h c ) ] E J c h c K ( ), J J E J max E (J\J) J 1 c {m} h J J J (J \J) J c {m} where n the last lne we use the nducton assumton (8) for J < m Fnally (10), (11) and (1) mly (9) and comlete the roof Remark The roof of Prooston 6 below wll use a verson of Corollary for nonnegatve random functons takng values n L r The nequalty s as follows: for > 1 there exsts K m,,r < such that (8 ) E ( h Km,,r max E J max E J c ) h ] Jc The roof s smlar to the revous ones and s omted: one takes (H ) as the startng ont of the nducton Fnally we come to the thrd ste, whch wll extend Hoffmann-Jørgensen s nequalty (H) for 1 If we want to use the nequaltes from Corollary to obtan boundedness of moments from stochastc boundedness of a sequence of U-statstcs, we need to relace the term corresondng to J = by the -th ower of a quantle of h For ths we use Paley- Zygmund s nequalty (eg, Kahane (1968) or de la Peña and Gné (1999)): f A s a nonnegatve random varable and 0 < r < <, then, for all 0 < λ < 1, (13) } Pr {A > λ A r (1 λ r ) A ] /( r) r, A )

9 Gné, Lata la and Znn 9 where A r = ( E A r) 1/r for 0 < r < Consder for nstance nequalty (8) It has the form EA B + K m(ea), > 1, wth A = h Then, ether B Km(EA), n whch case we have EA B, or B < Km(EA), n whch case we have EA Km(EA) and we can aly Paley-Zygmund s (13) wth λ = 1/ and r = 1 It gves {A > 1 } EA 1 Pr Hence, f we defne (14) t 0 = nf t 0 : Pr{A > t} we obtan EA t 0 So, n ether case, Also, by Markov s nequalty, We then have: (+1)/( 1) K /( 1) m EA B + 1+ K mt 0 1 (+1)/( 1) K /( 1) m 1 (+1)/( 1) K /( 1) m t 0 EA Theorem 3 Under the hyotheses of Prooston 1, there exst a unversal constants K m < such that, f t 0 s as defned by (14) for A = h, then 1 (4K m ) /( 1) t 0 max (15) ( ) E h (4K m ) { max J 1+ t 0 + J E J max ( ) ] E J ch c c ], ( ) ] } J E J max E J ch A smlar nequalty wth dfferent constants can be obtaned from (8 ) Ths s the most elaborate form we wll gve to our bounds for h 0 and > 1 The rght hand sde of (15) for m = becomes, dsregardng constants, ( E, h, ) C max E 1 max ( E h, ), E max ( ), E 1 h, (15 ) E max h,,, t 0 ]

10 10 Inequaltes for U-statstcs So, we get the -th moment of the double sum controlled by moments of artal maxma of condtonal exectatons lus a quantle The Gné-Znn (199) nequalty (for m = ), ( E, h, ) C max E max ( ) ], h, t 0, 1, s slghtly weaker n aearance than (15 ) (actually, we only ublshed the result for canoncal U-statstcs, but we aled t as well to nonnegatve varables, for whch the roof s the same: see, eg, Gné and Zhang (1996)) For alcatons of ths nequalty n the asymtotc theory of U-statstcs see Gné and Zhang (1996), Gné, Kwaeń, Lata la and Znn (1999) and de la Peña and Gné (1999) Remark The constants n the defnton of t 0 n (15) deend on, hence, so does t 0 Ths s not the case when m = 1 (as a consequence of the mroved Hoffmann-Jørgensen s nequalty of Kwaeń and Woyczyńsk -see, de la Peña and Gné (1999) 11-) But n most alcatons t does not matter whether the defnton of the quantle deends on Canoncal kernels, moments or order If ξ are centered and ndeendent and, then, by convexty and the Khnchn-Bonam nequalty (eg, de la Peña and Gné, 1999, 113), we have ) / E( ξ E ε ξ E ξ E ) ε ξ ( 1) E( / ξ /, (16) where ε are ndeendent dentcally dstrbuted Rademacher random varables, ndeendent from {ξ } Suose h s canoncal for the varables } gven n the revous subsecton, that s, suose {X () (17) E h(x (1) 1,, X (m) ) = 0 as for all = 1,, m, 1 1,, n Let ε () set be an ndeendent Rademacher array ndeendent of {X () }, and ε := ε (1) 1 ε (m) Then, recursve alcaton of nequalty (16) gves ( ) / m E h m E ε h E h m E ( ) / (18) ε h m ( 1) m/ E h Ths nequalty reduces estmaton of moments of canoncal U-statstcs to estmaton of moments of nonnegatve ones (and conversely), at least f constants are not an ssue Combned wth Prooston 1, t gves the

11 Gné, Lata la and Znn 11 analogue of Rosenthal s nequalty for centered varables and >, and f we aly t n conuncton wth Corollary, we obtan the followng nequalty: Prooston 4 If, for > and all {1,, n} m, h (X (1) 1,, X (m) ) s -ntegrable and E h (X (1) 1,, X (m) ) = 0 as for all = 1,, m, then ) ] / (19) m max K m ( E J max E J ch E h c ( ) ] / (m+ J )/ E J max E J ch for unversal constant K m < And, alyng Paley-Zygmund wth r =, we fnally have: Theorem 5 Let h be as n Prooston 4, and let > Then, there exst unversal constants K m < such that, f t 0 s defned as { } ( 3 ) ] /( ) 1 t 0 = nf t 0 : Pr h > t 4 ( K m m/), 1/( ) then 1 (4K m m/ ) /( ) t 0 max m max E (0) h { Km ( m/ ) t 0 + J J c E J max ( ) ] / E J ch c c ( ) ] } / (m+ J )/ E J max E J ch If, nstead of nequalty (), we wsh to obtan an analogue of nequalty ( ), that s, f we want to relace the constants at the rght hand sde of (19) by (K/ log ) m, then we cannot use Khnchn s nequalty and must roceed drectly wth an nducton as n Prooston 1 wth the followng change: we must consder the varables c h as takng values n L (J c ) and aly nequalty (15) n Kwaeń and Szulga (1991), whch gves Rosenthal s nequalty wth best constants for centered ndeendent random varables n Banach saces We sk the detals 3 Nonnegatve kernels, moments of order 1 It seems mossble to obtan nequaltes as smle as n the revous secton for ths case However, one can stll obtan nequaltes that may become useful when combned wth Paley-Zygmund Here s an analogue of Corollary for

12 1 Inequaltes for U-statstcs h 0 and 1 The method of roof s neffcent regardng constants as Hoffmann-Jørgensen s aled twce at each ste Hence, constants wll not be secfed Prooston 6 Let 0 < r < 1, m < and assume that the kernels h 0 have ntegrable -th owers Then (1) ( ( ) r ) ] /r ( ) max E J max E J c h E h c ( ( ) r ) /r K r,,m max E J max E J c h ], where K r,,m deends only on the arameters r,, m Note that all the terms n ths bound reresent a reducton n the number of sums excet for the term corresondng to J =, whch conssts of a ower of the r-th moment of a U-statstc of order m We wll deal later wth ths term by means of the Paley-Zygmund argument Proof The nequalty at the left sde of (1) follows from Hölder Inequalty (H r ) s ust the rght hand sde of nequalty (1) for m = 1 and we can roceed by nducton We stll use the notaton from Prooston 1 By the nducton hyothess we have () ( ) ( E h = Em E {1,,} K r,, J {1,,} {1,,} E J E m max c h ) E J c ( c h ) r ] /r Let us fx J {1,, m 1} and note that, for fxed (X () ) J, we have ( max E J c 1 J c h ) r := hm for sutably chosen ndeendent rv s h m n l (L r ) Therefore by (H r ), whch stll holds n ths sace (as the norm, restrcted to nonnegatve vectors, s monotone ncreasng), we have ( E J E m max E J c 1 J c h ) ] r /r = E E J m /r hm

13 (3) C,r E J E m max h m /r + Gné, Lata la and Znn 13 (E m ( = C,r E J {m} max (E J c {m} c h + E J (E m max E J c ( c ) ] /r hm ) r ) /r h ) r ) ] /r Now, to estmate the last term, we note that (4) K /r,1, J E J E m max E J c ( c h ) r ] /r ( ) ] r /r E J E J c {m} h ( E J max E J\J J 1 c {m} J J J \J J c {m} h ) r ] /r, whch follows by the verson of Corollary for L r ((8 ) for /r > 1) Now (), (3) and (4) comlete the nducton ste To deal wth the term corresondng to J = n Prooston 6 we aly Paley-Zygmund as above, but now wth r < relacng 1 < The concluson s: Theorem 7 There s a constant K r,,m such that for 0 < r < 1, m <, and h 0 wth ntegrable -th owers, we have 1 ( +1 K r,,m ) 1/( r) t 0 ( ( ) r ) ] /r E J max E J c h ( ) (5) E h K r,,m { J /r t 0 + J c ( ( ) r ) ] } /r E J max E J c h, c where t 0 = nf t : Pr { } h > t 1 ] (+1 K r,,m ) 1/( r)

14 14 Inequaltes for U-statstcs Hence, the -th moment of a U-statstc of order m can be estmated by artal moments of maxma (or sums) of condtonal moments of U- statstcs of lower order lus the -ower of a quantle of the orgnal U- statstc 4 Canoncal kernels and moments of order 1, or kernels h searately symmetrc n each of the coordnates and 0 < < 1 The canoncal case reduces to the ostve case by means of nequalty (18), as before The convexty art of nequalty (18) fals for < 1, but n ths case, f h s symmetrc searately n each of the coordnates, we can stll randomze by roducts of ndeendent Rademacher varables and recursve alcaton of Khnchn s nequalty stll reduces ths case to nonnegatve h We leave the resultng statements to the reader n order to avod reetton 5 Regular (undecouled) general U-statstcs If h (x) = h s (x s) for any ermutaton s of {1,, m} and h = 0 f has reeated ndces, and f the sequences {X () : = 1,, n} are ndeendent coes of each other, then the decoulng nequaltes of de la Peña and Montgomery-Smth (1995), together wth the decoulng nequalty for maxma n Htczenko (1988) n combnaton wth the revous nequaltes gve moment nequaltes for the generalzed U-statstcs h 1,, (X 1,, X m ) where {X } s a sequence of ndeendent random varables, at the cost of vastly ncreasng the numercal constants (see eg Gné and Znn (199) for a smlar alcaton of the decoulng nequaltes) We omt the resultng statements 6 Comarson wth revous results We have already noted, below the statement of Theorem 3, that the nequaltes there are better than the Hoffmann-Jørgensen tye nequaltes for U-statstcs n Gné and Znn (199) n that they reresent a decomoston nto smler quanttes Also, as mentoned n the Introducton, Ibrahmov and Sharakhmetov (1998, 1999) obtaned, excet for constants, Prooston 1 and ts analogue for canoncal kernels for m = and announced the result for general m; the fnal results n the resent artcle for > 1 n the nonnegatve case (Theorem 3) and for > n the canoncal case (Theorem 5), relacng some sums by maxma and lower moments by quantles, seem to be more useful As mentoned above, Corollary restrcted to m = recovers nequaltes (414) n Klass and Nowck (1997) The nequaltes n the last mentoned artcle for nonnegatve kernels, < 1 and m = (the nonconvex case, nequaltes (413) there) are dfferent from our nequaltes n Theorem 7 for m =, although they reresent a smlar level of decomoston of the -th moment of the U-statstc Bascally, the dfference s that they use nverses of truncated condtonal moments whereas we use

15 Gné, Lata la and Znn 15 nverses of tal robabltes together wth artal moments Ths can be better seen by comarng Hoffmann-Jørgensen, whch s Theorem 7 for m = 1, wth ther nequalty for m = 1 The result of Klass and Nowck (1997) can be descrbed as the teraton of an nequalty that follows from Hoffmann-Jørgensen, Paley-Zygmund ((13)) and (3), as follows Gven ξ, = 1,, n, nonnegatve, defne v 0 as { v 0 = su v 0 : ( ξ ) } (6) E v 1 1 or, what s the same, v 0 s the largest number satsfyng (7) v 0 = E ( ξ v 0 ) Then, the nequalty n queston s: Corollary 8 (Klass and Nowck, 1997, Cor 7) Let ξ, 1 = 1,, n, be ndeendent nonnegatve random varables Then, for all > 0, (8) E ( ξ ) E max ξ + v 0 Proof Snce E ( ξ δ 0 ) = Eξ I ξ<δ 0 + δ 0 Pr{ξ δ 0 } δ 0, t follows that δ 0 v 0 Therefore, f 1, nequalty (3) and the defnton of v 0 gve ) ( ) E( ξ E(ξ v 0 ) + Eξ I ξ >v 0 v 0 + E max ξ And f > 1, Hoffmann-Jørgensen ((H)) and the revous nequalty (wth = 1) gve ( ) ( E ξ < E ) ξ + E max ξ < v 0 + E max ξ For the reverse nequalty, f > 1, v 0 = ( E(ξ v 0 )) E ( ξ ) And f < 1, followng the roof of Lemma n Klass and Nowck (1997), we frst observe that Paley-Zygmund and the frst art of ths roof gve that for some unversal constant C, ( { Pr ξ v 0 > v } 0 1 E ) (ξ v 0 ) 4 (ξ ) = E( 1 v0 4 (ξ ) v 0 ) E( v 0 ) v 0 C 4 E max(ξ v 0 ) + v0 C 8 ;

16 16 Inequaltes for U-statstcs therefore, E( ξ ) E ( (ξ v 0 ) ) E ( (ξ v 0 )) I (ξ v 0)>v 0/ ] C 8 v 0 ( In fact, f we bound t 0 by t 0 E (ξ t 0 )) and aly the above roof to the varables ξ t 0, Hoffmann-Jørgensen gves the followng seemngly weaker nequalty: lettng ṽ 0 be the arameter v 0 for the smaller varables ξ t 0 (note ṽ 0 v 0 ), then ( ) ( ) E ξ E max ξ + ṽ 0 3 Imroved moment nequaltes and exonental nequaltes for m = The rght hand sde of nequalty (19) for m = 1 s ust E ξ K max E max ξ, /( ) / ] Eξ (31),, where ξ are ndeendent mean zero random varables These nequaltes were frst obtaned by Pnels (1994) Part of ther nterest le on the fact that they are bascally equvalent to Bernsten s nequalty u to constants Here s how (31) (for all ) mles Bernsten s nequalty u to constants Assume ξ A < for all, and set C = Eξ Then, (31) has the form E ξ K max A, / C ], Let = x ( x ) KeA KeC for any x for whch Then, by Markov s nequalty, (31) gves, for these values of x, K A x e f A / C { } Pr ξ > x K / C x e otherwse Hence, { } { Pr ξ > x e e = e ex x ( x ) } (3) KeA KeC for all x > 0 Smlarly, from the teraton (19) of the nequaltes (31) we can obtan exonental nequaltes for generalzed decouled U-statstcs

17 Gné, Lata la and Znn 17 of any order However, the nequaltes we obtan, whle better than the exstng ones, are not of the best knd, as we wll see below We llustrate ths comment by consderng the case m = In ths case, nequalty (19) s as follows: E h, K max ( (33),, Eh,) /, 3/ E 1 max 3/ E max ( ( /, E h,) /, E 1 h,) E max h, ], For bounded canoncal kernels h, we defne (34) A = max h,, C = Eh,,,, B = max E 1 h,(x (1), y), E h,(x, X () ) ] Then, we can roceed as n the deducton of (3) from (31), and easly obtan from (33) that there s a unversal constant K such that { } { Pr h, > x K ex 1 x ( x ) /3, ( x ) 1/ ] } (35) K mn C, B A, Ths nequalty also holds for regular canoncal U-statstcs by the decoulng nequaltes of de la Peña and Montgomery-Smth (1995) Inequalty (35) s better than the Bernsten tye nequalty n Arcones and Gné (1993) as t s better for x n A and the robablty s zero for x n A Inequalty (35) s subotmal for small values of x, for whch the exonent should be a constant tmes x, ust as for chaos varables of order (see Hanson and Wrght (1971), Ledoux and Talagrand (1991) and Lata la (1999)) Ths suggests that nequalty (9) s not of the best knd, and can be mroved Next we mrove the Rosenthal tye nequalty (9) for m = (that s, (33)) and deduce from t an exonental nequalty for canoncal U- statstcs of order two whch does detect the Gaussan orton of the tal robablty Frst we show how Talagrand s (1996) extenson of Prohorov s nequalty to emrcal rocesses, actually n Massart s (1999) verson, roduces an mroved Rosenthal s nequalty for emrcal rocesses Then, we wll use ths nequalty to estmate the terms resultng from condtonally alyng nequalty (31) to the U-statstc To descrbe Massart s verson of Talagrand s nequalty we must establsh the settng and defne some arameters Let Z be ndeendent random varables wth values n some measurable sace (T, T ), let F be a countable class of measurable real functons on T, and defne S := su f F f(z ), σ = su f F E(f(Z )), a := max su f F f(z )

18 18 Inequaltes for U-statstcs Then, (36) { Pr S E S + σ } 8x + 345ax e x for all x > 0 It follows easly from nequalty (36) that (37) E S K (E S ) + / σ + a ] for some unversal constant K < and all 1, n fact, nequalty (37) for all large enough and nequalty (36) for all x > 0 are equvalent u to constants (We do not lan to kee track of constants n the dervaton below and, therefore, we refran from secfyng a value for K n (37)) Prooston 31 Let {Z } be as above, let F be a countable class of functons such that Ef (Z ) < and Ef(Z ) = 0 for all Then, n the notaton from the revous aragrah, (38) E S K (E S ) + / σ + E max for all 1, where K s a unversal constant su f(z ) ] Proof Set F := su f F f and M := 8 3 E max F (Z ) Snce the varables f(z ) are centered, we can randomze by ndeendent Rademacher varables ε ndeendent of the Z varables (at the rce of ncreasng the value of the constant K) Set S := su f ε f(z ) Then, S su f ε f(z )I F (Z) M + su f f F ε f(z )I F (Z)>M := S1 + S, and notce that, snce ES 1 +1 E S (eg, Lemmas 16 and 143 n de la Peña and Gné, 1999), nequalty (37) gves ES 1 K (E S ) + / σ + M ] To estmate ES we aly the orgnal Hoffmann-Jørgensen nequalty (from eg, Ledoux and Talagrand (1991), (69) n age 156) to get ES 3( t 0 + E max F (Z ) ), where t 0 s any number such that Pr{S > t 0 } (8 3 ) 1 But the choce of M mles that we can take t 0 = 0 because Pr { S > 0 } = Pr { max F (Z ) > M } 1 8 3, rovng the rooston In what follows we wll assume, ust as above, that the kernels h,,, n, are comletely degenerate and defne (39) D = (h, ) L L := su { E, h, (X (1), X () )f (X (1) )g (X () )

19 : E Gné, Lata la and Znn 19 f (X (1) ) 1, E } g (X () ) 1 Theorem 3 There exsts a unversal constant K < such that, f h, are bounded canoncal kernels of two varables for the ndeendent random varables X (1), X (), = 1,, n, n N, then E h, (X (1), X () ) K /( Eh /,) + (h, ) L L (310) 1, n for all + 3/ E 1 max ] + E max h,,, ( / ( ) / ] E h,) + E max E 1 h, Inequalty (310) s strctly better than the rght hand sde nequalty n (9) for m =, that s, than (33) Proof Inequalty (31) aled condtonally on the varables X (1) (311) ( E h, K E 1 /, gves ( ) ] / E h, + ) E h, To bound the frst summand at the rght hand sde of (311) we frst notce that ( ) ] 1/ E h, = su E h, (X (1), X () )f (X () ) : E ] f (X () ) 1, where n fact, the su s taken only over a countable subset of mean zero vector functons (f 1,, f n ) dense n the unt ball of L (L(X () 1 ( )) 1/ L (L(X n () )) for the semnorm (f ) n = Ef (X () )) To see ths, frst aly dualty n l n and then n L (L(X () )) for each ] So we can aly (38) to Z = (h, ) n =1 wth f(z ) = E h,(x (1), X () )f (X () ) In ths case, the rght hand sde terms n (38) can be estmated as follows The frst term: ( ) ] (E S ) E S = E E h, = E h, = C,

20 0 Inequaltes for U-statstcs For the second we see that, snce, by the revous dualty argument, ( E 1 E h, (X (1), X () )f (X () )) (h, ) L L = D, t follows that σ D The thrd term: ] E max su f(z ) = E 1 max su E h, (X (1), X () )f (X () ) f E 1 max = E 1 max Thus, nequalty (38) gves (31) E f 1 ( ( / ) ) / E E h, (E su E h f 1, / (E h,) ) 1/ ( E K / C + D + 3/ E 1 max (E h, f ) / ] ) 1/ ] To estmate the second summand at the rght hand sde of (311), we aly (31) once more and obtan (313) E E 1 h, ( ) / K 3/ E E 1 h, + E ] h, Thus, to comlete the roof of the theorem t suffces to relace the sum n and the sum n, resectvely by maxma n and n, on the terms at the rght hand sde of ths nequalty But ths s an easy exercse of alcaton of nequalty (6) For comleteness sake, here t s Alyng (6) wth α = 3 and / nstead of, the frst term at the rght of (313) bounds as: ( ) / 3/ E E 1 h, ( ) 3/E ( ) / 1+3/ (1 + (/) 3 ) max E 1 h, + C ], whch roduces the converson of the sum nto a maxmum wthout ncreasng the order of the multlcatve constant n front of C The second term n (313) requres two stes Frst, we aly (6) for / and α = 4,,

21 Gné, Lata la and Znn 1 condtonally on {X (1) }: E h, (314), ( ) E ( ) ] / +1 (1 + (/) 4 )E 1 max h, + E h, We aly (6) wth resect to E 1, for / and α = 0, to the second term at the rght hand sde of (314) and we obtan the bound ( ) / +3 (1 + (/) 4 ) E 1 max E h, + C ], whch s n terms of some of the quanttes aearng at the rght hand sde of (310) and wth coeffcents of lower order As for the frst term at the rght of (314), we aly (6) wth resect to E 1, agan for / and α = 4, and get t bounded by 4+ (1 + (/) 4 ) ( ) E max, h, + E ( ) ] / E 1 max h, Here the frst term concdes wth the last one n (310), and the second s domnated by ( / K E E 1 h,)] Alyng nequalty (R 1 ) wth resect to E ths s n turn domnated by K ( ) /E ( / E 1 h,) + K C, and the frst summand has alredy been handled above (frst term at the rght of (313)) Collectng terms we obtan nequalty (310) Theorem 3 mles the followng moment nequalty and exonental bound for bounded kernels Theorem 33 There exst unversal constants K < and L < such that, f h, are bounded canoncal kernels of two varables for the ndeendent random varables X (1), X (), = 1,, n, and f A, B, C, D are as defned n (34) and (39), then (315) E 1, n h, (X (1), X () ) K / C + D + 3/ B + A ]

22 Inequaltes for U-statstcs for all and, equvalently, { } Pr h, (X (1), X () ) x (316), n for all x > 0 L ex 1L ( x mn C, x D, x/3 B A )] x1/, /3 1/ The moment nequalty s mmedate from Theorem 3 and the equvalence wth the exonental nequalty follows ust lke (3) follows from (31) n one drecton, and, n the other, by ntegraton of tal robabltes Next we comment on the exonental nequalty For comarson uroses, let h, (X (1), X () ) = g g x, wth g, g ndeendent standard normal In ths case, C = { x, and D = su,, u v x, : u 1, } v 1 and the Gaussan chaos nequaltes of Hanson and Wrght (1971) and Lata la (1999) yeld the exstence of unversal constants 0 < k < K < such that { } Pr h, K(Cx 1/ + Dx) e x and,, { } Pr h, k(cx 1/ + Dx) k e x By the central lmt theorem for canoncal U-statstcs, ths mles that the coeffcents of x and x n (316) are correct (excet for K) It s natural to have terms n smaller owers of x n (316) eg, by comarson wth Bernsten s nequalty for sums of ndeendent random varables In fact, the term n x 1/ cannot be avoded, at least u to logarthmc factors To see ths, consder the roduct V of two ndeendent centered Posson varables wth arameter 1, whch s the lmt n law of V n =, n X(n) where X (n) and Y (n) are centered Bernoull random varables wth arameter Y (n) = 1/n; then, for large x, the tal robabltes of V are of the order of ex ( x 1/ log x), and therefore, so are those of V n for large n Also, note that the term n x /3 n the exonent corresonds, u to logarthmc factors, to the tal robabltes of the roduct of two ndeendent random varables, one normal and the other centered Posson If X, Y, X (1), X () are d, h, = h for all, and h s comletely degenerate, then the arameters defned by (34) and (38) become: A = h, B = n ( E Y h (x, Y ) + E X h (X, y) ), C = n Eh

23 and Gné, Lata la and Znn 3 { } D = n su Eh(X, Y )f(x)g(y ) : Ef (X) 1, Eg (Y ) 1 := n h L L, where h L L s the norm of the oerator of L (L(X)) wth kernel h Then, nequaltes (315) and (316) become: Corollary 34 Under the above assumtons, there exst unversal constants K <, L < such that, for all n N and, E h(x (1), X () ) K / n (Eh ) / + n h L L (317) and (318), n { Pr, n + 3/ n /( E Y h + E X h ) / + h h(x (1), X () ) x n h L L, } x K ex 1K ( x mn n Eh, x /3 n( EY h + E X h ) ] 1/3, x 1/ h 1/ )] Inequalty (318) rovdes an analogue of Bernsten s nequalty for degenerate U-statstcs of order : note that nequaltes (315), (316), (317) and (318) can all be undecouled usng the result of de la Peña and Montgomery-Smth s (1995) It should also be noted that ths exonental nequalty for canoncal U-statstcs s strong enough to mly the suffcency art of the law of the terated logarthm for these obects: ths can be seen by alyng t to the kernels h n n Stes 7 and 8 of the roof of Theorem 31 n Gné, Kwaeń, Lata la and Znn (1999) (and usng some of the comutatons there for the arameters C to D) Nether nequalty (35) nor any of the revously ublshed nequaltes for U-statstcs can do ths Acknowledgement We thank Stanslaw Kwaeń for several useful conversatons References ] Arcones, M and Gné, E (1993) Lmt theorems for U-rocesses Ann Probab de la Peña, V and Gné, E (1999) Decoulng: From Deendence to Indeendence Srnger-Verlag, New York de la Peña, V and Montgomery Smth, S (1995) Decoulng nequaltes for the tal robabltes of multvarate U-statstcs Ann Probab

24 4 Inequaltes for U-statstcs Fgel, T; Htczenko, P; Johnson, WB; Schechtman, G; and Znn, J (1997) Extremal roertes of Rademacher functons wth alcatons to the Khntchne and Rosenthal nequaltes Trans Amer Math Soc Gné, E; Kwaeń, S; Lata la, R; and Znn, J (1999) The LIL for canoncal U-statstcs of order two To aear Gné, E and Zhang, C-H (1996) On ntegrablty n the LIL for degenerate U-statstcs J Theoret Probab Gné, E and Znn, J (1983) Central lmt theorems and weak laws of large numbers n certan Banach saces Zets Wahrsch v Geb Gné, E and Znn, J (199) On Hoffmann-Jørgensen s nequalty for U-rocesses Probablty n Banach Saces Brkhäuser, Boston Hanson, D L and Wrght, F T (1971) A bound on tal robabltes for quadratc forms n ndeendent random varables Ann Math Statst Htczenko, P (1988) Comarson of moments for tangent sequences of random varables Probab Th Rel Felds Johnson, W B; Schechtman, G; and Znn, J (1985) Best constants n moment nequaltes for lnear combnatons of ndeendent and exchangeable random varables Ann Probab Ibragmov, R and Sharakhmetov, Sh (1998) Exact bounds on the moments of symmetrc statstcs In: Abstracts of the 7-th Vlnus Conference on Probablty Theory and Mathematcal Statstcs/ nd Euroean Meetng of Statstcans, Vlnus Ibragmov, R and Sharakhmetov, Sh (1999) Analogues of Khntchne, Marcnkewcz-Zygmund and Rosenthal nequaltes for symmetrc statstcs Scand J Statst Kahane, J-P (1968) Some Random Seres of Functons Heath, Lexngton, Massachusetts Klass, M and Nowck, K (1997) Order of magntude bounds for exectatons of functons of nonnegatve random blnear forms and generalzed U-statstcs Ann Probab Kwaeń, S and Szulga, J (1991) Hyercontracton methods n moment nequaltes for seres of ndeendent random varables n normed saces Ann Probab Kwaeń, S and Woyczyńsk, W (199) Random Seres and Stochastc Integrals: Sngle and Multle Brkhäuser, Boston Lata la, R (1997) Estmaton of moments of sums of ndeendent random varables Ann Probab Lata la, R (1999) Tals and moment estmates for some tye of chaos Studa Math Lata la, R and Znn, J (1999) Necessary and suffcent condtons for the strong law of large numbers for U-statstcs Ann Probab, to aear

25 Gné, Lata la and Znn 5 Ledoux, M and Talagrand, M (1991) Probablty n Banach Saces: Isoermetry and Processes Srnger, New York Massart, P (1999) About the constants n Talagrand s concentraton nequaltes for emrcal rocesses Ann Probab, to aear Pnels, I (1994) Otmum bounds for the dstrbutons of martngales n Banach saces Ann Probab Talagrand, M (1996) New concentraton nequaltes n roduct saces Invent Math Utev, S A (1985) Extremal roblems n moment nequaltes In: Lmt Theorems n Probablty Theory, Trudy Inst Math, Novosbrsk, (n Russan) Evarst Gné Rafa l Lata la Deartment of Mathematcs Insttute of Mathematcs and Deartment of Statstcs Warsaw Unversty Unversty of Connectcut Banacha Storrs, CT Warszawa USA Poland gne@uconnvmuconnedu rlatala@mmuwedul Joel Znn Deartment of Mathematcs Texas A&M Unversty College Staton, TX znn@mathtamuedu