Pointwise and uniform asymptotics of the Vervaat error process

Transcription

1 Poitwise ad uiform asymptotics of the Vervaat error process Edre Csáki 1 Alfréd Réyi Istitute of Mathematics, Hugaria Academy of Scieces, Budapest, P.O.B. 127, H-1364, Hugary. csaki@reyi.hu Miklós Csörgő 2 School of Mathematics ad Statistics, Carleto Uiversity, Ottawa, Otario, Caada K1S 5B6. mcsorgo@math.carleto.ca Atóia Földes 3 City Uiversity of New York, 28 Victory BLVD., State Islad, New York 1314, U.S.A. afoldes@gc.cuy.edu Zha Shi Laboratoire de Probabilités UMR 7599, Uiversité Paris VI, 4 Place Jussieu, F Paris Cedex 5, Frace. zha@proba.jussieu.fr Ričardas Zitikis 4 Departmet of Statistics, Uiversity of Maitoba, 327 Machray Hall, Wiipeg, Maitoba, Caada R3T 2N2. Ricardas Zitikis@umaitoba.ca Abstract. It is well kow that, asymptotically, the appropriately ormalized uiform Vervaat process, i.e., the itegrated uiform Bahadur Kiefer process properly ormalized, behaves like the square of the uiform empirical process. We give a complete descriptio of the strog ad weak asymptotic behaviour i -orm of this represetatio of the Vervaat process ad, likewise, we study also its poitwise asymptotic behaviour. Keywords. Empirical process, quatile process, Bahadur Kiefer process, Vervaat process, Vervaat error process, Kiefer process, Browia bridge, Wieer process, strog approximatio, law of the iterated logarithm, covergece i distributio. 2 Mathematics Subject Classificatio. Primary 6F15; Secodary 6F5, 6F17. 1 Research ported by the Hugaria Natioal Foudatio for Scietific Research, Grat No. T Research ported by a NSERC Caada Grat at Carleto Uiversity, Ottawa. 3 Research ported by a PSC CUNY grat, No Research ported by a grat of the Uiversity of Maitoba, Wiipeg, ad by the NSERC Caada Collaborative Grat at the Laboratory for Research i Statistics ad Probability, Carleto Uiversity ad Uiversity of Ottawa, Ottawa. 1

2 1 Itroductio ad prelimiary results Let U 1, U 2,... be idepedet copies of a radom variable U uiformly distributed over the iterval [, 1]. Let F t := 1 1{U k t}, t 1, k=1 deote the empirical distributio fuctio based o U 1, U 2,..., U, where 1 is the idicator fuctio. Let F 1 be the left-cotiuous iverse of F. We deote the empirical ad quatile processes over the iterval [, 1] by respectively. The sum α t := F t t, t 1, β t := F 1 t t, t 1, R t := α t + β t, t 1, of the empirical ad quatile processes is kow i the literature as the Bahadur Kiefer process cf. Bahadur, 1966, Kiefer, 1967, 197. The Bahadur Kiefer process ejoys some remarkable asymptotic properties, which are of iterest i statistical quatile data aalysis cf., e.g., Csörgő, 1983, Shorack ad Weller, We summarize the most relevat results of Kiefer 1967, 197 i this regard i the followig theorem. Throughout the paper, we use the otatio log x := log maxx, e ad log 2 x := log log x. Theorem A For every fixed t, 1, we have lim 1/4 R t d t1 t 1/4 N Ñ,, 1.1 1/4 R t 1/4 25/4 = t1 t a.s., 1.2 log 2 3/4 3 3/4 where N ad Ñ are idepedet stadard ormal variables ad d deotes covergece i distributio. Also, lim 1/4 log R = 1 a.s., 1.3 α where f := t 1 ft deotes the uiform -orm of f. Theorem A is due to Kiefer 1967, 197, except for 1.3 for which he oly proved the covergece i probability. The upper boud for the almost sure covergece i 1.3 was proved by Shorack 1982, ad the lower boud by Deheuvels ad Maso 199. For a review of these results ad for further developmets alog these lies we refer to Deheuvels ad Maso 1992, Eimahl 1996, Csörgő ad Szyszkowicz Via usig the usual ad the other laws of the iterated logarithm for α, 1.3 immediately implies lim 1/4 log log 2 1/4 R = 2 1/4 a.s., 1.4 lim if 1/4 log log 2 1/4 R = π 8 1/4 a.s., 1.5 2

3 while a direct applicatio of 1.3 together with the weak covergece of α to a Browia bridge B gives 1/4 log R d B,. 1.6 Nevertheless, the followig result, which oe ca immediately coclude also by combiig 1.1 with 1.6, is true, ad it was first formulated ad proved directly by Vervaat 1972b. Theorem B Vervaat, 1972b The statemet a R d Y,, caot hold true i the space D[, 1] edowed with the Skorohod topology for ay sequece {a } of positive real umbers ad ay o-degeerate radom elemet Y of D[, 1]. I view of Theorems A ad B ow, it is of iterest to see the asymptotic behaviour of the Bahadur Kiefer process possibly i other orms as well. I this regard we quote Theorem C Csörgő ad Shi, 1998 For ay p [2,, we have where lim 1/4 R p α p/2 = c p a.s., 1.7 c p := E N p 1/p = 2 Γp + π 1/p, 1.8 N stads as before for a stadard ormal variable, ad f p := 1 ft p dt 1/p, the Lp orm of f. I particular, 1.7 yields L p versios of the laws of the iterated logarithm LIL s for the Bahadur Kiefer process R. We also ote that 1.7 combied with the weak covergece of α to a Browia bridge B implies that, for p [2,, 1/4 R p d c p B q,. This, i combiatio with 1.6, yields Theorem B agai. Vervaat s 1972b proof of Theorem B was based, i a most crucial ad elegat way, o the followig itegrated Bahadur Kiefer process I t := t R sds, t 1. Cocerig the latter process, he established the weak covergece of V t := 2 I t 1.9 to B 2, the square of a Browia bridge, as well as a fuctioal LIL for V, via provig the followig theorem for a discussio of related details we refer to Csörgő ad Zitikis, 1999a, b. 3

4 Theorem D Vervaat, 1972a, b We have I particular, i the space C[, 1], lim 2 1 V α 2 = a.s. 1.1 lim α 2 = i probability V d B 2, For use of termiology, we call the process V of 1.9 the uiform Vervaat process. For further refereces ad elaboratio o this termiology we refer to Sectio 1 of Zitikis As a cosequece of 1.12, Vervaat 1972b cocluded Theorem B. We also ote that 1.1 yields the LIL for V, see Corollary 1.1 Vervaat 1972a, b i Csörgő ad Zitikis 1999a, b. 2 The Vervaat error process, mai results Bahadur 1966 itroduced R as the remaider term i the represetatio β = α + R of the quatile process β i terms of the empirical process α. As we have see i Theorems A ad C, the remaider term R, i.e., the Bahadur Kiefer process, is asymptotically smaller tha the mai term α, i.e., the empirical process, i both the L p ad -orm topologies. I a similar vei, we ca thik of the process Q t := V t α 2 t, t 1, 2.1 that appears i both statemets 1.1 ad 1.11 of Theorem D as the remaider term Q i the followig represetatio V = α 2 + Q 2.2 of the uiform Vervaat process V i terms of the square of the empirical process. It is well-kow cf. Zitikis, 1998, for details ad refereces that the remaider term Q i 2.2 is asymptotically smaller tha the mai term α 2. Thus, just like i the case of R, oe may like to kow how small the remaider term Q is. I view of Theorems A ad C, oe suspects that there should be substatial differeces betwee the asymptotic poitwise, - ad L p -orms behaviour of the process Q. Ideed, Csörgő ad Zitikis 1999a established the followig strog covergece result for Q p. Theorem E Csörgő ad Zitikis, 1999a For ay p [1,, we have lim 1/4 Q p α 3p/2 = 1 c 3/2 p a.s.,

5 where c p is defied i 1.8. For a compariso of this result to that of Theorem C, as well as for that of their cosequeces, we refer to Csörgő ad Zitikis 1999a, who have also cojectured that i -orm the aalogue statemet of 2.3 should be of the followig form: lim b 1/4 Q = c a.s., 2.4 α 3/2 where b is a slowly varyig fuctio covergig to ad c is a positive costat. Oe of the aims of this expositio is to prove that this cojecture is true with b = log. I additio, we also study the poitwise behaviour of the Vervaat error process Q. We summarize our results i the followig theorem, which parallels Theorem A of Kiefer 1967, 197 cocerig the process R. Theorem 2.1 For every fixed t, 1, we have lim 1/4 Q t d 4/3 t1 t 3/4 N Ñ 3/2,, 2.5 1/4 Q t log 2 5/4 = t1 t 3/4 211/4 3 1/4 5 5/4 a.s., 2.6 where N ad Ñ are idepedet stadard ormal variables. Also, lim 1/4 log Q α = 3/2 4/3 a.s. 2.7 As a cosequece of this theorem, as well as that of Theorem E combied with 2.7, we have the followig corollary, which cofirms Cojecture 2.1 of Csörgő ad Zitikis 1999a. Corollary 2.1 The statemet a Q d Y,, caot hold true i the space D[, 1] for ay sequece {a } of positive real umbers ad for ay o-degeerate radom elemet Y of the space D[, 1]. Aother cosequece of 2.7 is the followig corollary. Corollary 2.2 We have lim 1/4 log log 2 3/4 Q = 21/4 3 a.s., 2.8 lim if 1/4 log log 2 3/4 π 3/2 Q = 3 2 5/4 a.s., 2.9 1/4 log Q d 4/3 B 3/2,, 2.1 where B is a stadard Browia bridge. 5

6 We ote that 2.8 ad 2.9 follow from 2.7 by meas of the usual ad the other LIL s for α. As to 2.1, it results from a direct applicatio of 2.7 together with the weak covergece of α to the Browia bridge B. Remark 2.1 I the literature we also fid geeral forms of the Vervaat process that are based o radom variables X 1, X 2,... replacig the uiform [,1] radom variables U 1, U 2,... I particular, such a geeral Vervaat process first appeared ad was put to good use i Csörgő ad Zitikis We refer to Zitikis 1998 for a detailed survey o this subject. For related though rather differet limit theorems for the geeral Vervaat process, we refer to Csörgő ad Zitikis It is obvious that the results of this paper ca be geeralized i such a way that they would cover geeral forms of the Vervaat process as well. However, a solutio of this problem uder reasoably optimal assumptios may costitute a rather challegig mathematical task which is defiitely ot withi the scope of the preset paper. For a recet review of Vervaat ad Vervaat error processes we refer to Csörgő ad Zitikis 1999b. 3 The Vervaat error process i terms of a Kiefer process We itroduce some two-parameter Gaussia processes. Let {W x, y, x, y } be a Wieer Browia sheet, i.e., a two-parameter Gaussia process with EW x, y = ad covariace fuctio EW x 1, y 1 W x 2, y 2 = x 1 x 2 y 1 y 2. Next we defie a Kiefer process {Kx, y, x 1, y } by Kx, y := W x, y xw 1, y, where W x, y is a Wieer sheet. A Kiefer process Kx, y ca be characterized as a mea zero Gaussia process with covariace fuctio EKx 1, y 1 Kx 2, y 2 = x 1 x 2 x 1 x 2 y 1 y 2. Remark 3.1 I this paper we defie, without loss of geerality, all Kiefer processes Kx, y ad Browia bridges Bx to be equal to zero if x < or x > 1. Cocerig the uiform empirical process α, Komlós et al. followig fudametal embeddig theorem established the Theorem F Komlós, Major ad Tusády, 1975 O a suitable probability space, the uiform empirical process {α k x, x 1, k = 1, 2,... } ad a Kiefer process {Kx, k, x 1, k = 1, 2,... } ca be so costructed that, as, max 1 k x 1 α k x Kx, k = O log 2 Combiig 1.4 with Theorem F, we arrive at cf. Csörgő ad Révész, 1975: a.s. 6

7 Propositio A O the probability space of Theorem F for the uiform quatile process {β k x, x 1, k = 1, 2,... } with the Kiefer process {Kx, k, x 1, k = 1, 2,... } of Theorem F, as, we have max 1 k x 1 β k x + Kx, k = O log log 2 1/4 1/4 a.s. 3.1 We ote i passig cf. Deheuvels, 1998 ad refereces therei that the almost sure rate of covergece i 3.1 caot be improved eve whe approximatig the uiform quatile process by ay other Kiefer process. For further use we quote the followig two results of A.H.C. Cha 1977 cf. Theorem ad Theorem S , respectively, i Csörgő ad Révész, Theorem G Cha, 1977 Let K, be a Kiefer process, ad let {h } be a o-icreasig sequece of positive umbers such that lim log 1 h / log 2 =. The lim t 1 h Kt + s, Kt, = 1 a.s. s h 2h log1/h Theorem H Cha, 1977 Let < ε T < 1, < a 2 T T be fuctios of T such that ε T ad a T /T are o-icreasig ad a T is o-decreasig. Defie K x 1, x 2 ], t = Kx 2, t Kx 1, t x 1 < x 2 1. The almost surely, where lim T = lim T t T a T t T a T β T = x 1 ε T β T Kx, x + s], t + a T Kx, x + s], t s ε T β T Kx, x + ε T ], t + a T Kx, x + ε T ], t = 1, x 1 ε T 2a T ε T 1 ε T log T + log ε T a 2 T. T If i additio, lim T log T ε T a T / log 2 T =, the lim ca be replaced by lim. T T The mai result of this sectio is the followig strog approximatio of the Vervaat error process Q t defied i 2.1 via a Kiefer process. Theorem 3.1 O the probability space of Theorem F, usig the there costructed Kiefer process K,, Q ca be approximated as follows. As we have Q t Z t = O 3/8 log 3/4 log 2 5/8 a.s., 3.2 <t<1 where {Z t, < t < 1, = 1, 2,... } is defied by Kt, Z t := 2 1 K t s Kt,, Kt, ds. 3.3 The proof of this theorem is based o the ext two lemmas, which are of iterest o their ow. 7

8 Lemma 3.1 Let The A t := 2 t ad, cosequetly, as, F 1 t α u α t du, < t < 1, = 1, 2, Q t = A t R 2 t, 3.5 Q t A t = O log log 2 a.s. 3.6 <t<1 Proof. We have the followig easy-to-check represetatio V t = α tβ t + A t α t + β tβ t for all t, 1 ad = 1, 2,... cf., Vervaat, 1972a, Shorack ad Weller, 1986, Zitikis, By 2.1, Q t = V t αt 2 = A t Rt, 2 yieldig 3.5. The idetity 3.6 follows from 3.5 ad 1.4. Lemma 3.2 O the probability space of Theorem F, usig the there costructed Kiefer process K,, the stochastic process A t of Lemma 3.1 ca be approximated such that whe, A t Z t = O 3/8 log 3/4 log 2 3/8 a.s., <t<1 where Z t is as i Theorem 3.1. Proof. By a chage of variables u = t st F 1 t = t + s β t i 3.4, we have 1 A t = 2β t α t + s β t α t ds. The usual LIL for β cofirms that β = Olog 2 almost surely whe. Therefore, by Theorem F, whe, 1 A t = 2 β t K t + s β t, Kt, ds + O log 2 log 2, a.s., 3.7 uiformly i t, 1. For all t, s, 1, we have K t + s β t, Kt, K t s, := Ku + v, Ku, u 1 Ku + v, Ku,, v h 8

9 where u = t s Kt, β t Kt,, v = s + ad, o accout of Propositio A, h = O 3/4 log log 2 1/4, almost surely. Thus, accordig to Theorem G, almost surely, whe, K t + s β t, Kt, K t s, = O 1/8 log 3/4 log 2 1/8, uiformly i t, s, 1. Pluggig this ito 3.7 ad usig agai the LIL for β, we arrive at: 1 A t = 2 Kt, β t K t s, Kt, ds Accordig to Propositio A, +O 3/8 log 3/4 log 2 5/8 a.s Kt, β t = 2 + O log log 2 1/4, a.s., 3.9 almost surely ad uiformly i t, 1. O the other had, applyig Theorem G to the itegrad i 3.9 with h = O log 2, we obtai: 1 Kt, K t s, Kt, ds = O 1/4 log log 2 1/4, almost surely ad uiformly i t, 1. Pluggig this ad 3.9 ito 3.8 yields that almost surely, whe, log log2 A t = Z t + O + O 3/8 log 3/4 log 2 5/8, 3/4 uiformly i t, 1. This yields Lemma 3.2. Proof of Theorem 3.1. Follows from Lemmas 3.1 ad 3.2. We metio that the process Z t was put to use i Csörgő ad Zitikis 1999a i their study of Q t i L p orm. Moreover, they also remarked that their cojecture as stated i 2.4 is equivalet to statig it i terms of Z istead of Q. Likewise, i view of Theorem 3.1 ow, the proof of Theorem 2.1 ca, ad will, be based o provig the statemets with the process Z t replacig Q t i all of them. The latter goal i tur will be achieved via usig the ext simple, though essetial, observatio that is borrowed from Eimahl Propositio B Let {Kx, y, x 1, y } be a Kiefer process. For ay fixed u < v 1, the process { } Ku + v ux, y xkv, y 1 xku, y, x [, 1], y v u 9

10 is a Kiefer process, idepedet of {Ks, y, s [, u], y } ad {Ks, y, s [v, 1], y }. Based o Theorem 3.1 ad this crucial observatio, the respective proofs of the poitwise statemets of ad the proof of the uiform property as i 2.7 of the process Q will take differet routes. Hece, our ext Sectio 4 is devoted to provig , while Sectio 5 will be o establishig the almost sure ratio statemet of Poitwise behaviour of the Vervaat error process This sectio is devoted to the proof of 2.5 ad 2.6 of Theorem 2.1, cocerig the poitwise asymptotics of Q t. We first establish a strog approximatio of Z t of 3.3 for ay fixed t, 1 by a process i, which is a itegral of a Wieer sheet i its first parameter over a radom iterval that is idepedet of this Wieer sheet i had. Lemma 4.1 Give Z t as i 3.3, the for ay fixed t, 1 oe ca defie a Wieer sheet W, such that, as, we have Kt, / Z t = 2 W y, dy + O log log 2 where {W y,, y, = 1, 2,... } is idepedet of {Kt,, = 1, 2,... }. a.s., 4.1 Proof. Usig Propositio B with u =, v = t, ad writig 1 x istead of x, we see that K 1x, := Kt1 x, 1 xkt, t, x 1, 4.2 is a Kiefer process, idepedet of Kt,. Likewise, lettig u = t, v = 1, we see that K 2x, := Kt + 1 tx, 1 xkt, 1 t, x 1, is also a Kiefer process, idepedet of Kt,. Moreover, K 1, ad K 2, are idepedet. Cosequetly, the compoets of the vector are idepedet processes. Defiig x via Kt,, K 1,, K 2, 4.3 Kt, 1 xt = t s Kt, t + 1 tx = t s if Kt, >, if Kt, <, i the itegral i the defiitio of Z t of 3.3, we arrive at the followig represetatio of the latter process for each fixed t: Z t = I 1 1{Kt, > } I 2 1{Kt, < }, 4.4 1

11 where Kt,/t I 1 := 2t Kt1 x, Kt, dx, I 2 := 21 t Kt,/1 t Kt + 1 tx, Kt, dx. Cosiderig I 1, ad rememberig that this is the case whe Kt, >, usig the defiitio 4.2 of K 1,, we obtai: almost surely as, I 1 = 2t 3/2 Kt,/t = 2t 3/2 Kt,/t = 2t 3/2 Kt,/t Kt,/t K1x, dx 2tKt, x dx K1x, tkt, 3 dx 2 t 2 K1x, log2 3/2 dx + O, 4.5 the last lie followig from the LIL for the Kiefer process with fixed t. Sice K 1, is a Kiefer process idepedet of Kt,, we ca use the represetatio K 1x, = W 1 x, xw 1 1,, where W 1, is a Wieer sheet, idepedet of Kt,. This idepedece property will be crucial i our use later o. Observe that Kt,/t K 1x, dx = Kt,/t W 1 x, dx K2 t, W 1 1, 2t 2 2. By the LIL for the Kiefer process K ad the Wieer process W1 1,, we have, whe, K 2 t, W1 1, = O 3/2 log 2 3/2 almost surely. Goig back to 4.5, we obtai: as, almost surely Kt,/t I 1 = 2t 3/2 W1 log2 3/2 x, dx + O Kt,/ = 2 t W1 y t, dy + O log2 3/ Similarly, i the case Kt, <, we ca show that as, for I 2 of 4.4 we have, almost surely, Kt,/ I 2 = 2 1 t W2 y log2 3/2, dy + O, t where, just like W 1, of 4.6, the Wieer sheet W 2, of 4.7 is also idepedet of Kt, cf Combiig 4.6 ad 4.7 with 4.4 yields that, for each fixed t, 1, as, Kt, / Z t = 2 W y, dy + O 11 log2 log a.s.,

12 where W y, := t W1 y t, 1{Kt, > } 1 t W2 y, 1{Kt, < }. 1 t Sice W, is a Wieer sheet, idepedet of Kt,, this yields Lemma 4.1. The rest of this sectio is devoted to the proof of 2.5 ad 2.6 i Theorem 2.1. For the sake of clarity, they are proved separately. 3 Proof of 2.5. For each fixed ad T, T T 3/2 W y, dy is a stadard ormal variable. Thus, by coditioig, if T is a radom variable idepedet of W,, the 3 T T 3/2 W y, dy is a stadard ormal variable, idepedet of T. I view of the idepedece of Kt, ad W,, we have, for each fixed ad t, takig T := Kt, / = Kt, / 2 W y, dy 4 t1 t 3 3/4 T 1/4 t1 t 3/2 3 T 3/2 T W y, dy = d 4 3 t1 t 3/4 1/4 Ñ 3/2 N, where = d deotes idetity i distributio, ad N ad Ñ are idepedet stadard ormal radom variables. This, i light of 3.2 ad 4.1, yields 2.5. Proof of 2.6. Fix agai t, 1. Defie W t := Kt,, 1, t1 t W t x, y := t1 t 1/4 W x t1 t, y, x, y. Clearly, W t is a Wieer process restricted o N, ad W t, is a Wieer sheet, idepedet of {Kt,, = 1, 2,... }, thus of W t. We also observe that Kt, / W y, dy = t1 t 3/4 2 log 2 5/4 1/4 Wt /2 log 2 Wt u 2 log 2, du. 1/4 2 log 2 3/4 By Theorem 1.1 of Deheuvels ad Maso 1992, as, the set of limit poits of is almost surely equal to W t 2 log 2, W t u 2 log 2,, u [, 1] 1/4 2 log 2 3/4 D := {c, f : f absolutely cotiuous with respect to the Lebesgue measure, c, 1, f =, c 2 + c f u 2 du 1}

13 Cosequetly, with probability oe, lim 1/4 Kt, / log 2 5/4 W y, dy = 2 5/4 t1 t 3/4 c,f D c fu du. We ow determie the value of the expressio o the right had side. Itegratig by parts, usig the Cauchy Schwarz iequality ad 4.8, we get, for each c, f D, c c fu du = c c u 2 du c uf u du c c f u du c 2. 3 Sice c 3 1 c /2 /5 5/2 for ay c [ 1, 1], this yields Choosig c fu du 2 3 1/ c,f D 5 5/4 c = 3/5, fu = 2 3 1/4 u 51/4 5 1/ /4 u2, it is see that i 4.9 we have, i fact, a equality. Accordigly, with probability oe, lim 1/4 Kt, / W y, dy = t1 t 3/4 27/4 3 1/4. log 2 5/4 5 5/4 This yields 2.6 i view of 4.1 ad Uiform behaviour of the Vervaat error process This sectio is devoted to establishig the uiform property 2.7 i Theorem 2.1. I view of Theorems 3.1 ad F it suffices to show that lim lim if log log Z K, 3/2 Z K, 3/2 where Z := t 1 Z t ad K, := t 1 Kt,. Proof of 5.1. Recall that cf. 3.3, by defiitio, Kt, Z t = 2 = 2 Kt,/ 1 K t s 4 3 a.s., a.s., 5.2 Kt,, Kt, ds Kt z, Kt, dz

14 Let N = N := log 2 4, 5.4 ad let t i = t i := i/n for i N. Defie, for u [, 1], Bi,u := N K t i + u N, ukt i+1, 1 ukt i,. It follows from Propositio B that for each fixed, {B i,, i =, 1,..., N 1} are idepedet Browia bridges, ad idepedet of {Kt i,, i =, 1,..., N 1}. Let t [t i, t i+1 ]. The Kti,/ Kt,/ Z t = 2 + Kt z, Kt, dz. Kt i,/ By Theorem G, whe, Kt, Kt i, log max = O i N 1 t i t t i+1 N O the other had, by the LIL for the Kiefer process ad Theorem G, Kt z Kt, = O 1/4 log log 2 1/4, uiformly i i N 1, t [t i, t i+1 ] ad z [Kt i, /, Kt, /]. Therefore, Kt,/ log log2 1/4 Kt z, Kt, dz = O a.s. Kt i,/ N 1/4 As a cosequece, for t [t i, t i+1 ], we have, almost surely whe, Kti,/ log log2 1/4 Z t = 2 Kt z, Kt, dz + O, 5.5 N 1/4 where O is uiform i i =, 1,..., N 1. Let i be such that Kt i,. If t [t i, t i+1 ], the t = t i + v/n for some v [, 1]. Accordigly, by writig A i := N Kt i, /, Kti,/ = N 3/2 Kt z, Kt, dz B i,v + y B i,v dy Kt i+1, Kt i, N Sice A i y dy = N 2 K 2 t i, /2 2, a applicatio of the LIL for the Kiefer process ad Theorem G yields that with probability oe, Kti,/ Kt z, Kt, dz = N 3/2 a.s. a.s., y dy. N Bi,v + y Bi,v log log dy + O 2,

15 uiformly i i =, 1,..., N 1. Note that B i, ca be represeted as B i,u := W i,u + uw i,1, where for each fixed {W i,, i =, 1,..., N 1} are idepedet Wieer processes which are also idepedet of {Kt i,, i =, 1,..., N 1}. Hece, almost surely, = = B i,v + y B i,v dy W i,v + y W i,v dy + W W i,v + y W i,v dy + O i,1 y dy N 2 log log 2, 5.7 the last equality followig from the LIL for the Kiefer process ad the fact that almost surely, i N 1 Wi,1 = Olog. This last fact ca be easily checked usig the usual estimate for Gaussia tails ad the Borel Catelli lemma, regardless of the depedecy structure of the stadard ormal variables {Wi,1, i =, 1,..., N 1; = 1, 2,... }. Puttig 5.5, 5.6 ad 5.7 together, ad takig ito accout the defiitio of N i 5.4, we obtai: Z t = 2 N 3/2 W i,v + y W i,v dy + O log, 5.8 1/4 log 2 for t = t i + v/n, v 1, Kt i,. If Kt i, >, the by itroducig the Kiefer process K t, := K1 t,, we ca write Kti,/ 2 Kt z, Kt, dz = 2 Kti,/ K 1 t + z, K 1 t, dz, ad, similarly to the argumet leadig to 5.8, we ca fid idepedet Wieer processes {Wi,, i =, 1,..., N 1} which are also idepedet of {Kt i,, i =, 1,..., N 1}, such that almost surely, Z t = 2 N 3/2 W i,v + y W i,v dy + O log, 5.9 1/4 log 2 for t = t i + v/n, v 1, Kt i, >. Combiig 5.8 ad 5.9, we see that with probability oe, for t = t i + v/n, v [, 1], Z t = 2 N 3/2 log W i, v + y W i, v dy + O, 5.1 1/4 log 2 where W i, := W i, 1{Kt i, } + W i, 1{Kt i, > }, 15

16 i =, 1,..., N 1, are Wieer processes, idepedet of {Kt i,, i =, 1,..., N 1}. Note that we do ot claim that the Wieer processes {W i,, i =, 1,..., N 1} are idepedet betwee themselves. For each, we split {, 1,..., N 1} ito two radom parts: J 1 = J 1 := {i : Kt i, log 2 1, i N 1}, J 2 = J 2 := {i : Kt i, > log 2 1, i N 1}. If i J 1, the applyig the LIL for the Kiefer process ad Theorem G implies that Kti,/ Kt z, Kt, dz = O log 1/4 log 2 3/2 a.s. uiformly i i J 1. I view of 5.5, we obtai: whe, max i J 1 t [t i,t i+1 ] Z t = O log 1/4 log 2 3/2 a.s For i J 2, we cosider the variables X i, := 3 A 3/2 i v 1 Accordig to 5.1, almost surely whe, max i J 2 t [t i,t i+1 ] Z t 2 K, 3/2 3 W i, v + y W i, v dy. max i J 2 X i, + O log, 1/4 log 2 which, combied with 5.11, yields Z 2 K, 3/2 3 max i J 2 X i, + O log 1/4 log 2 a.s We ow show that the variables X i,, i J 2 have Gaussia-like tails. To this ed, we fix A > ad itroduce the mea zero Gaussia process Xv := 3 A W v + y W v dy, v [, 1]. A 3/2 It is straightforward to compute its covariace: EXuXv = { 1 3 v u + v u 3 if v u A, 2A 2A 3 if v u > A, 5.13 ad therefore E Xv Xu 2 = { 3 v u A v u 3 if v u A, A 3 2 if v u > A

17 Accordig to a well-kow iequality of Ferique 1964, for ay x >, P Xv x σ ϕp s2 ds 4p 2 e s2 /2 ds, v 1 1 x where p 2 is arbitrary, ad σ ad ϕ are such that EXv 2 σ 2, E Xv Xu 2 ϕ 2 v u. I view of 5.13 ad 5.14, we ca choose σ = 1 ad ϕh = 3h/A. Sice 3 ϕp s2 ds e s log p/2 2 3 ds =, A pa log p this leads to: P 1 Xv x v pa log p 4p 2 e s2 /2 ds Recall that W i, is idepedet of Kt i,. Therefore, applyig 5.15 to W = W i, ad A = A i for i J 2 we obtai: P X i, x log 2 3 4p 2 e s2 /2 ds, p log p where, with N as i 5.4, we used the fact that A i log 2 6 for all i J 2. Let ε, 1. We choose p = ε 2 log 2 6 ad = ε such that for all, 8 3 log 2 3 / p log p ε. Thus for ay x > ad, which yields PX i, 1 + εx 4log 2 12 ε 4 P max X i, 1 + εx 4Nlog 2 12 e x2 /2 i J 2 ε 4 x Takig x := 1 + εlog, ad we obtai: P max X i, 1 + ε 2 log i J 2 x e s2 /2 ds 4log 2 12 x x ε 4 e x2/2 x, 4 log 2 8 e x2 /2. ε 4 x 4log 2 8 ε εlog ε. Let k = k 2/ε. By the Borel Catelli lemma ad 5.12, almost surely whe k, Z k 21 + ε2 log k K, k 3/2 log k + O k 1/4 k log 2 k Let k < k+1. Note that k+1 k = O 1 ε/2 k, k. By 5.3, Kt,/ Z t Z k t 2 k,,t z dz Kt,/ +2 Kt z, k Kt, k dz Kt, k / k, 17

18 where k,,t z := Kt z, Kt z, k Kt, + Kt, k. Accordig to Theorem H ad the LIL for the Kiefer process, whe k, k,,t z = O 1 ε/4 k log k log 2 k 1/4 a.s., uiformly i t, 1, z [, Kt, /] ad k < k+1. Thus, by the LIL for the Kiefer process, Kt,/ k,,t z dz = O 1+ε/4 k log k log 2 k 3/4 a.s., uiformly i t, 1 ad k < k+1. O the other had, by the LIL for the Kiefer process ad Theorem G, Kt z, k Kt, k = O 1/4 k log k log 2 k 1/4 a.s., uiformly i t, 1, z [Kt, k / k, Kt, /] ad k < k+1. Sice xxxx I eed a referece for this; Edre has a trick to prove it by meas of Theorem H, he is goig to isert a proof here Kt, Kt, k = O ε/4 k log k a.s., 5.17 t 1 uiformly i k < k+1, it follows that Kt, max Kt, k k < k+1 t [,1] k = O ε/4 k log k a.s. Therefore, almost surely whe k, Kt,/ Kt, k / k Kt z, k Kt, k dz = O 1+ε/4 k log k log 2 k 1/4, uiformly i t [, 1] ad k < k+1. We have therefore proved that max Z Z k = O 1+ε/4 k < k log k log 2 k 1/4, a.s. k+1 Goig back to 5.16 ad takig 5.17 ito accout, we obtai, whe, Z 21 + ε2 log log K, 3/2 + O 3 1/4 log 2 = 1 + o ε2 log K, 3/2, 3 We used i the last lie the other LIL for the Kiefer process Kuelbs, 1979, Mogulskii 1979: K, 1 = O log2 a.s. a.s The proof of 5.1 is complete. 18

19 Proof of 5.2. Let N ad t i, i =, 1,..., N be as before cf Combiig Theorem G with 1 log Kt, = O a.s t 1 cf. Csáki ad Shi, 1998, we coclude max i N 1 Kt i, lim t 1 Kt, = 1 a.s. 5.2 Let t + = t + ω, be such that Kt +, = max i N 1 Kt i, ad cosider the set of idices I := {i : t + h t i t + + h }, where h = log 4. Fix ε, 1. For i I, we obtai from Theorem G, 5.19 ad 5.2 that whe is sufficietly large, Kt i, Kt +, O h log1/h 1 εkt +, >, 5.21 Hece, for i I, we ca apply 5.9 to t = t i ad v = to see that whe, Z t i = 2 NKti,/ log W N i,y dy + O 3/2 1/4 log 2 2 log = 3 Kt i, 3/2 Y i, + O, /4 log 2 where, for each, {Wi,, i =, 1,..., N 1} are idepedet Wieer processes which are also idepedet of {Kt i,, i =, 1,..., N 1}, ad Y i, := 3 3/2 N 3/2 1 Kt i, 3/2 NKti,/ W i,y dy, i I. For each, {Y i,, i I } are idepedet stadard ormal variables. Sice #I N/log 4, we have, by the usual estimate for Gaussia tails, P max Y i, < 1 ε log i I 2 N/log 4 + o ε π log 1 ε/2 2 + o1 N/log 4 exp 1 ε π log 1 ε/2 = exp 2 + o1 ε/2, 1 ε π log 9/2 log 2 4 which sums. By the Borel Catelli lemma, almost surely for all large, max i I Y i, 1 ε log. Pluggig this ito 5.22 ad 5.21 yields that, almost surely, Z max Z t i i I 21 ε3/2 log log Kt + 3, 3/2 + O 1/4 log 2 21 ε2 log 3/2 log Kt, + O. 3 t 1 1/4 log 2 19

20 We used 5.2 i the last iequality. This also holds true for if t 1 Kt, i place of t 1 Kt,, by meas of a similar argumet with Kt +, replaced by Kt, = mi i N 1 Kt i,. Cosequetly, Z 21 ε2 log log K, 3/2 + O 3 1/4 log 2 a.s. I light of 5.18, this yields 5.2. Refereces [1] Bahadur, R.R A ote o quatiles i large samples. A. Math. Statist [2] Cha, A.H.C O the Icremets of Multiparameter Gaussia Processes. Ph.D. Thesis, Carleto Uiversity. [3] Csáki, E. ad Shi, Z Some limif results for two-parameter processes. Stoch. Process. Appl [4] Csörgő, M Quatile Processes with Statistical Applicatios. SIAM, Philadelphia. [5] Csörgő, M. ad Révész, P Some otes o the empirical distributio fuctio ad the quatile process. I: Coll. Math. Soc. J. Bolyai 11 Limit Theorems of Probability Theory, Keszthely, Hugary, pp North Hollad, Amsterdam. [6] Csörgő, M. ad Révész, P Strog Approximatios i Probability ad Statistics. Academic Press, New York. [7] Csörgő, M. ad Shi, Z A L p -view of the Bahadur Kiefer theorem. Tech. Report No. 324 of the Laboratory for Research i Statistics ad Probability, Carleto Uiversity ad Uiversity of Ottawa, Ottawa. [8] Csörgő, M. ad Szyszkowicz, B Sequetial quatile ad Bahadur Kiefer processes. I: Order Statistics: Theory & Methods eds. N. Balakrisha ad C.R. Rao pp Elsevier Sciece B.V., Amsterdam. [9] Csörgő, M. ad Zitikis, R Strasse s LIL for the Lorez curve. J. Multivariate Aalysis [1] Csörgő, M. ad Zitikis, R O the geeral Bahadur Kiefer, quatile, ad Vervaat processes: old ad ew. Tech. Report No. 323 of the Laboratory for Research i Statistics ad Probability, Carleto Uiversity ad Uiversity of Ottawa, Ottawa. To appear i: Bolyai Society Mathematical Studies: A Volume i Hoour of Pál Révész. [11] Csörgő, M. ad Zitikis, R. 1999a. The Vervaat process i L p -spaces. Tech. Report No. 328 of the Laboratory for Research i Statistics ad Probability, Carleto Uiversity ad Uiversity of Ottawa, Ottawa. [12] Csörgő, M. ad Zitikis, R. 1999b. O the Vervaat ad Vervaat-error processes. Acta Applicadae Mathematicae [13] Deheuvels, P O the local oscillatios of empirical ad quatile processes. I: Asymptotic Methods i Probability ad Statistics a Volume i Hoour of Miklós Csörgő ed. B. Szyszkowicz pp Elsevier Sciece B.V., Amsterdam. 2

21 [14] Deheuvels, P. ad Maso, D.M Bahadur Kiefer-type processes. A. Probab [15] Deheuvels, P. ad Maso, D.M A fuctioal LIL approach to poitwise Bahadur Kiefer theorems. I: Probability i Baach Spaces R.M. Dudley et al., eds. pp Birkhäuser, Bosto. [16] Eimahl, J.H.J A short ad elemetary proof of the mai Bahadur Kiefer theorem. A. Probab [17] Ferique, X Cotiuité des processus gaussies. C. R. Acad. Sci. Paris [18] Kiefer, J O Bahadur s represetatio of sample quatiles, A. Math. Statist [19] Kiefer, J Deviatios betwee the sample quatile process ad the sample df. I: Noparametric Techiques i Statistical Iferece ed. M.L. Puri pp Cambridge Uiversity Press, Cambridge. [2] Komlós, J., Major, P. ad Tusády, G A approximatio of partial sums of idepedet R.V. s ad the sample DF. I. Z. Wahrsch. verw. Gebiete [21] Kuelbs, J Rates of growth for Baach space valued idepedet icremet processes. Lecture Notes i Mathematics 79 pp Spriger, New York. [22] Mogulskii, A.A O the law of the iterated logarithm i Chug s form for fuctioal spaces. Th. Probab. Appl [23] Shorack, G.R Kiefer s theorem via the Hugaria costructio. Z. Wahrsch. Verw. Gebiete [24] Shorack, G.R. ad Weller, J.A Empirical Processes with Applicatios to Statistics. Wiley, New York. [25] Vervaat, W. 1972a. Fuctioal cetral limit theorems for processes with positive drift ad their iverses. Z. Wahrsch. Verw. Gebiete [26] Vervaat, W. 1972b. Success Epochs i Beroulli Trials: with Applicatios to Number Theory. Mathematical Cetre Tracts 42 secod editio i Matematisch Cetrum, Amsterdam. [27] Zitikis, R The Vervaat process. I: Asymptotic Methods i Probability ad Statistics a Volume i Hoour of Miklós Csörgő ed. B. Szyszkowicz pp Elsevier Sciece B.V., Amsterdam. 21