Clustering rates and Chung type functional laws of the iterated logarithm for empirical and quantile processes 2 1 0 Davit Varron 2 Laboratoire de Mathématiques de Besançon, UMR CNRS6623, n Universitéde Franche-Comté a e-mail: [email protected] J 6 Abstract: FollowingtheworksofBerthet[2,3],wefirstobtainexactclus- 2 teringratesinthefunctionallawoftheiterated logarithmfortheuniform empiricalandquantileprocessesandfortheirincrements.Inasecondtime, weobtainfunctionalChung-type limitlawsforthelocalempiricalprocess ] T foraclassoftargetfunctionsontheborderoftheStrassenset. S AMS 2000 subject classifications:Primary62G20, 62G30. h. Keywords and phrases: Empirical processes, Strassen laws of the iter- t atedlogarithm,Clusteringrates, Chung-Mogulskiilimitlaws. a m [ 1. Introduction 1 Definetheuniformempiricalprocessbyα (t):=n1/2(F (t) t),whereF (t):= v n n n − 1 n−1♯ i 1,...,n , Ui t , t [0,1], and (Ui)n≥1 are independent, identi- ∈ { } ≤ ∈ 2 cally distributed (i.i.d) random variables uniformly distributed on [0,1]. Define 5 the q(cid:8)uantile process by (cid:9) 5 . 1 β (t)=n1/2 F−1(t) t , t [0,1], 0 n n − ∈ 2 (cid:16) (cid:17) 1 where Fn−1(t):=inf{u: Fn(u)≥t}. In a metric space (E,d) we write un H v: wheneverunisrelativelycompactwithlimitsetH(see,e.g.,[17]).Thetwoabove mentioned processes have been extensively investigated in the literature (see, i X e.g., [20] and [24] and the references therein). In a pioneering work, Finkelstein r [10] has established the functional law of the iterated logarithm (FLIL) for α . n a Namely, the author showed that, writing log u = log(log(u e)) and b = 2 ∨ n 2log n, we have : 2 α n , (1.1) p a.s. 2 b S n in the metric space (B[0,1], ), where B[0,1] stands for the set of bounded || · || functions on [0,1] and is the sup-norm over [0,1]. The set in (1.1) is 2 || · || S given by := f(t) , f(1)=0 , (1.2) 2 1 S ∈S n o 1 imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 2 where · 1 := f B[0,1], f′ Borel, f():= f′(t)dt, f′2(t)dt 1 . (1.3) 1 S ∈ ∃ · ≤ (cid:26) Z Z (cid:27) 0 0 Note that (resp. ) is the unit ball of the reproducing kernel Hilbert space 2 1 S S of the Brownian bridge (resp. of the Wiener process) on [0,1]. In the spirit of [10],Mason[17]hasobtainedthefollowingFLILforthelocalempiricalprocess: α (a ) n n· . (1.4) a.s. 1 √a b S n n Here,a isasequenceofconstantssatisfyinga 0, na andna /log n n n ↓ n ↑∞ n 2 → .DeheuvelsandMason[8]haveestablishedarelateduniformfunctionallimit ∞ law for the following collections of random trajectories. α (t+a ) α (t) n n n Θ := · − , t [0,1 a ] . n n 2a log(1/a ) ∈ − n n n o They showed that, with propbability one : lim sup inf g f =0, n n→∞ gn∈Θnf∈S1 || − || lim sup inf g f =0, (1.5) n n→∞ f∈S1gn∈Θn || − || where a is a sequence of constants fulfilling a 0, na , na /logn n n n n ↓ ↑ ∞ → , log(1/a )/log n . Berthet [2] refined (1.5) under slightly stronger ∞ n 2 → ∞ conditions imposed upon a . Making use of sharp upper bounds for Gaussian n measures due to Talagrand [22], he proved that for any ǫ > ǫ (where ǫ is a 0 0 universal constant), we have almost surely for all n large enough : Θ +ǫlog(1/a )−2/3 . (1.6) n 1 n 0 ⊂S B Here := f B[0,1] : f 1 . The first aim of the present article is to 0 B { ∈ || ||≤ } showthatthe techniquesemployedinthe just-mentionedresultcanbe adapted to some other random objects than that used for that given in (1.6) (see The- orems 1 and 2 in the sequel). Results of this kind are usually called clustering rates. Another related problem is finding rates of convergence of such random sequences to a specified function belonging to the cluster set. Such results are known under the name of functional Chung-type limit laws. We now focus on thelocalempiricalprocessα (a ),wherea 0asn .TheworksofCsáki n n n · ↓ →∞ [5], de Acosta[1], Grill[12], GornandLifshits [11], and BerthetandLifshits [4] onsmallballprobabilitiesforWienerprocessesprovidesomecrucialtoolstoes- tablishsuchlimitlawsfor(α (a )) ,astheseareexpectedtoasymptotically n n n≥1 · mimic their gaussian analogues (see Mason [17]). Along this line, Deheuvels [6] established Chung-type limit laws for (α (a )) , by showing that, if a is a n n n≥1 n · imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 3 sequence of constants satisfying na , a 0 and na /(log n)3 , we n ↑ ∞ n ↓ n 2 → ∞ 1 have, almost surely, for each f satisfying f 2 := f′2(t)dt<1 : ∈S1 || ||H 0 R α (a ) π n n liminf (log n) · f = . n→∞ 2 √anbn − 4 1 f H (cid:12)(cid:12) (cid:12)(cid:12) −|| || (cid:12)(cid:12) (cid:12)(cid:12) The proof of this theorem rel(cid:12)i(cid:12)es on strong(cid:12)a(cid:12)ppropximation methods in combi- nation with the results of de Acosta [1]. The latter provides useful exponential bounds for W P f ǫ , T − ≤ (cid:16)(cid:12)(cid:12) (cid:12)(cid:12) (cid:17) with a small ǫ > 0 and a large T(cid:12)(cid:12). Here,(cid:12)W(cid:12) is a Wiener process on [0,1] and (cid:12)(cid:12) (cid:12)(cid:12) f satisfies f 2 < 1. The study of related probabilities when f = 1 has || ||H || ||H requireddifferent arguments.In [12], roughestimates are given. In [11] and [4], some exact rates are given, but only for functions with first derivatives having a variation either bounded or locally infinite. The sets of all functions of this type are called bv and liv respectively. In the present paper, we shall make S1 S1 use of the latter results to extend the work of Deheuvels [6] to the case where f bv liv. The remainder of our paper is organized as follows. Our main ∈ S1 ∪S1 resultsarestatedin 2,Theorems1,2and3.In 3,theproofsofthesetheorems § § are provided. 2. Main Results Our first result gives clustering rates in Finkelstein’s FLIL [10]. Theorem 1. There exists a universal constant ǫ >0 such that, for any choice 0 of ǫ>ǫ we have almost surely, for all large n 0 α n +ǫ(log n)−2/3 , (2.1) (2log n)1/2 ∈ S2 2 B0 2 β n +ǫ(log n)−2/3 . (2.2) (2log n)1/2 ∈ S2 2 B0 2 Remark 2.0.1. The uniform Bahadur-Kiefer representation (see [13]) asserts that, almost surely : limsupn1/4(logn)−1/2(log n)−1/4 α +β =2−1/4, 2 || n n || n→∞ from where (2.2) is readily implied by (2.1). Our second theorem concerns the FLIL for local increments of the empirical process. Theorem 2. Let a be positive real numbers satisfying, as n , n →∞ na n na , , a 0. (2.3) n ↑∞ (log n)7/3 →∞ n ↓ 2 imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 4 Then there exists a universal constant ǫ >0 such that, for any choice of ǫ>ǫ 1 1 we have almost surely, for all large n, α (a ) n n· +ǫ(log n)−2/3 . (2.4) 2a log n ∈S1 2 B0 n 2 If moreover na /(logpn)11/3 then we have, almost surely, ultimately as n 2 → ∞ n , →∞ β (a ) n n· +ǫ(log n)−2/3 . (2.5) 2a log n ∈S1 2 B0 n 2 Remark 2.0.2. Wepshall use the fact (see e.g. [9], Theorem 5) that, under (2.3), we have almost surely limsup(n/a )1/4(log n)−1/4(2log n+log(na ))−1/2 α (a )+β (a ) 2−1/4, n 2 2 n || n n· n n· ||≤ n→∞ from where (2.5) is implied by (2.4) after straightforward computations. Inordertostate ourlastresult,weneedto givesomedefinitions.Recallthat 1 f bv wheneverf′hasaderivativewithboundedvariationand f′2(t)dt=1. ∈S1 0 ResultsonsmallballprobabilitiesforaWienerprocesswhenf R bv havebeen ∈S1 established by Gorn and Lifshits [11]. For such a function f, we shall write (L) := L2/3, L > 0 and we denote by χ the constant which is the unique f f ∇ solutionofequation(3.1)in[11] (we referto the just mentionedpaperfor more 1 details).Thecasewheref liv (i.e.where f′2(t)dt=1andthederivativeof ∈S1 0 f′ admits a version with locally infinite variaRtion) has been treated by Berthet and Lifshits [4]. For such a function f, we set χ :=1 and we denote by (L) f f ∇ the uniquesolutionofequation(2.1)in[3].Ourthirdresultis statedasfollows. Theorem 3. Let f bv liv be arbitrary and let a be a sequence of real ∈ S1 ∪S1 n numbers satisfying, as n , →∞ na , a 0, a log n 0, (2.6) n ↑∞ n ↓ n 2 → na n lim = . (2.7) n→∞ log n 2(log n) ∞ 2 ∇f 2 Then we have, almost surely : α (a ) n n liminf (log n) · f =χ . n→∞ ∇f 2 2a log n − f (cid:12)(cid:12) n 2 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) Remark 2.0.3. The conditions (2.(cid:12)6(cid:12))pand (2.7) impos(cid:12)e(cid:12)d upon a turn out to be n the best possible with respect to the methods usedin the proof of Theorem 3. The latter combines poissonization techniques with strong approximation arguments. Deheuvels andLifshits [7]andShmileva[19]have provided newtoolstoestimate probabilities of shifted small balls for a Poisson process without making use of imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 5 strong approximation techniques. These results show up to be powerful enough to investigate Chung-Mogulskii limit laws for α (a .) without making use of n n strong approximation techniques, and thus relaxing condition (2.6). However, the just-mentioned results do not cover the case where f liv. ∈S1 3. Proofs 3.1. Proof of Theorem 1 Selectan ǫ>0 and considerthe sequence ǫ :=ǫ(log n)−2/3. The main toolto n 2 achieveourgoalisthe exponentialinequalitystatedinthe followingfact,which follows directly from Talagrand[21]. Recall that is the unit ball for . 0 B ||·|| Fact 3.1. Let B be a Brownian bridge. There exists three constants K, L and 0 u >0 such that, for any 0<u<u and c>0, we have : 0 0 L cu c2 0 P B / c +u Kexp . (3.1) ∈ S2 B0 ≤ u2 − 2 − 2 (cid:0) (cid:1) (cid:16) (cid:17) Let W be a Wiener process on [0,1]. There exist two constants u and L such 1 1 that, for any 0<u<u and c>0, we have 1 L cu c2 1 P W / c +u exp . (3.2) ∈ S1 B0 ≤ u2 − 2 − 2 (cid:0) (cid:1) (cid:16) (cid:17) In the proof of Theorem 1, we will make use of blocking techniques (see, e.g., [8] and [2]). For any real umber a, set [a] as the unique integer q fulfilling q a<q+1, and set ≤ n := exp kexp (logk)1/6 , k 1. k − ≥ h (cid:16) (cid:16) (cid:17)(cid:17)i Set N := n ,...,n 1 for k 5. Given an integer n 1, we set k(n) k k k+1 { − } ≥ ≥ as the unique integer k such that n N . We shall first study the following k ∈ sequence of functions g :=(n )−1/2b−1 H , k =k(n), n k+1 nk+1 n with H (t):=n(F (t) t) and b :=(2log n)1/2. Let p and q be two conju- n n − n 2 1 1 gates numbers (such that 1/p +1/q =1 ) with 1<p < . Set, for k 1, 1 1 1 ∞ ≥ 1 1 m := min P H H ǫ . p1,k n∈Nk (cid:18)(nk+1)1/2bnk+1 || nk+1 − n ||≤ p1 nk+1(cid:19) A standard blocking argument based upon Ottaviani’s inequality (see, e.g., [8], Lemma 3.4) yields 1 P H / +ǫ (n )1/2b n ∈S2 nk+1B0 (cid:18)n[∈Nkn k+1 nk+1 o(cid:19) 1 1 1 P H / + ǫ . ≤ m (n )1/2b nk+1 ∈S2 q nk+1B0 p1,k (cid:18) k+1 nk+1 1 (cid:19) imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 6 Letk be integerandselectn N .Bythe Dvoretsky-Kiefer-Wolfowitzinequal- k ∈ ity (see, e.g., [23]) we have : 1 1 P H H ǫ (n )1/2b || nk+1 − n ||≥ p nk+1 (cid:18) k+1 nk+1 1 (cid:19) P α 1 ǫ 1 1/2b ≤ || nk+1−n ||≥ p nk+1 1 nk nk+1 (cid:18) 1 − nk+1 (cid:19) (cid:0) (cid:1) 4ǫ2log (n )−1/3 3exp 2 k for large enough k, ≤ − p2 1 nk (cid:16) 1 − nk+1 (cid:17) (cid:0) (cid:1) whence m 1/2 for all large k by routine analysis. Now let p ,q > 1 be p1,k ≥ 2 2 two conjugate numbers.For k 1 we have,B denoting a Brownianbridge, ≥ nk+1 α 1 1 P nk+1 / + ǫ P α B ǫ b b ∈S2 q nk+1B0 ≤ || nk+1 − nk+1 ||≥ p q nk+1 nk+1 (cid:18) nk+1 1 (cid:19) (cid:18) 2 1 (cid:19) 1 +P B / b + ǫ b nk+1 ∈ nk+1S2 q q nk+1 nk+1B0 (cid:18) 2 1 (cid:19) :=PKMT +PTal. k k Making use of the Komlós-Major-Tusnàdy approximation (see, e.g., [14])), we can choose a sequence (B ) satisfying, for some universal constants C ,C nk k≥1 2 3 and for all k large enough, 1 PKMT C exp C (n )1/2 ǫ b . k ≤ 2 − 3 k+1 2p q nk+1 nk+1 2 1 (cid:16) (cid:17) Onthe other hand,by applying assertion(3.1) ofFact 3.1we have,forall large k, ǫ L (q q )2 PTal Kexp 0 1 2 (log n )1/3 log n . k ≤ − q q − 2ǫ2 2 k+1 − 2 k+1 1 2 h (cid:16) (cid:17) i Routine analysis shows that both PKMT and PTal are sumable in k for any k k choice of ǫ>(L /2)1/3 =:ǫ , provided that q ,q are chosen close enough to 1. 0 0 1 2 Now an application of (1.1) in combination with elementary properties of the sequence (n ) shows that, almost surely, as n , k k≥1 →∞ g b−1α =o((log n)−2/3). || n− n n || 2 3.2. Proof of Theorem 2 Recallthat b :=(2log n)1/2, n 1.Letp ,q >1 be twoconjugate numbers. n 2 ≥ 1 1 Set, for k 1 : ≥ 1 1 m := min P H (a ) H (a ) ǫ . p1,k n∈Nk (nk+1ank+1)1/2bnk+1 || n nk+1· − nk+1 nk+1· ||≤ p1 nk+1 (cid:16) (cid:17) imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 7 The same blocking argument as in §3.1 yields H (a ) P n nk+1· / +ǫ (n a )1/2b ∈S1 nk+1B0 (cid:18)n∈[Nkn k+1 nk+1 nk+1 o(cid:19) 1 H (a ) 1 P nk+1 nk+1· / + ǫ . ≤ m (n a )1/2b ∈S1 q nk+1B0 p1,k k+1 nk+1 nk+1 1 (cid:16) (cid:17) Now, for any integer k 5 and n N , we have k ≥ ∈ 1 1 P H (a ) H (a ) ǫ b √nk+1 || nk+1 nk+1· − n nk+1· ||≥ p1 nk+1 nk+1 (cid:16) (cid:17) P sup |αnk+1−n(t)| 1 ǫ b nk+1ank+1 1/2 . ≤ 1 t ≥ p nk+1 nk+1 n n (cid:16)t≤ank+1 − 1 (cid:0) k+1− k(cid:1) (cid:17) Itiswellknown(see,e.g.,[20],Proposition1,p.133)thatforeachn,theprocess (1 t)−1α (t) is a martingale in t. The Doob-Kolmogorovinequality yields : n − p2(1 a )(1 nk )(logn )1/3 1 m 1 − nk+1 − nk+1 k+1 . − p1,k ≤ 2ǫ2 Hence for all k large enough we have m 1/2. Now set for each integer p1,k ≥ n 1, ≥ ηn Π (t):=n−1/2 1 t , t [0,1], n {Ui≤t}− ∈ (cid:16)Xi=1 (cid:17) where η is a Poiesson variable with expectation n, which is independent of n (U ) . Let Π denote a standard centered Poisson process on R+ and let W i i≥1 be a Wiener process that we assume to be constructed on the same underlying probability speace as Π. Notice that Π () and n−1/2Π(n) are equal in distri- n · · bution as processes on [0,1]. Now let p ,q > 1 be two conjugate numbers. 2 2 By making use of Poeissonization techeniques (see, e.g.e, [8], Lemma 2.1 or [25], Proposition2.1foramoregeneralform)weseethat,forallsufficientlylargek : α (a ) 1 P nk+1 nk+1· / + ǫ (cid:18) a1n/k2+1bnk+1 ∈S1 q1 nk+1B0(cid:19) Π (a ) 1 2P nk+1 nk+1· / + ǫ ≤ (cid:18)ea1n/k2+1bnk+1 ∈S1 q1 nk+1B0(cid:19) Π(n a ) 1 =2P k+1 nk+1· / + ǫ (2n a log n )1/2 ∈S1 q nk+1B0 (cid:18) k+1 nk+1 2 k+1 1 (cid:19) e 1 2P W(n a ) Π(n a ) (n a )1/2b ǫ ≤ || k+1 nk+1· − k+1 nk+1· ||≥ q p k+1 nk+1 nk+1 nk+1 1 2 (cid:16) (cid:17) W(n a ) 1 +2P k+1 nk+1· e / + ǫ (n a )1/2b ∈S1 q q nk+1 k+1 nk+1 nk+1 1 2 (cid:16) (cid:17) :=PKMT +PTal. k k imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 8 Now,makinguseofthestrongapproximationtheoremofKomlós-Major-Tusnàdy [15], we can assume that the process W involved in the former expression sat- isfies, for some universal constants C ,C ,C >0, and for all T >0, z >0, 1 2 3 P Π(T ) W() z+C logT C exp C z . (3.3) 1 2 2 || · − · ||≥ ≤ − (cid:0) (cid:17) (cid:0) (cid:1) Notice that, as ke : →∞ (n a )1/2b ǫ k+1 nk+1 nk+1 nk+1 . log(n a ) →∞ k+1 nk+1 Thus, we have, ultimately as k , →∞ ǫC PKMT C exp 3 (n a )1/2(log n )−1/6 . (3.4) k ≤ 2 − √2q p k+1 nk+1 2 k+1 1 2 (cid:16) (cid:17) Recalling the assumption na /(log n)7/3 we see that PKMT is sumable n 2 → ∞ k in k. Now, making use of assertion (3.2) of Fact 3.1 we have, for all large k, 1 PTal = P W / b + ǫ b k ∈ nk+1S1 q q nk+1 nk+1B0 1 2 (cid:16) (cid:17) ǫ L (q q )2 exp 1 1 2 (log n )1/3 log n . ≤ − q q − 2ǫ2 2 k+1 − 2 k+1 (cid:18) (cid:16) 1 2 (cid:17) (cid:19) Now if ǫ > (L /2)1/3 =: ǫ and if q ,q > 1 are chose sufficiently small, then 1 1 1 2 PTal is sumable in k. By the Borel-Cantelli lemma, we see that for any ǫ > ǫ k 1 we have almost surely, for all large n, g +ǫ , n ∈S1 nk+1B0 where g := (n a )−1/2b−1 H (a ), n N . To conclude the proof n k+1 nk+1 nk+1 n nk+1· ∈ k of Theorem 2, it remains to control the distance between a−1/2b−1α (a ) and n n n n· g , which is the purpose of the following lemma. n Lemma 3.1. We have almost surely : α (a ) limsup(log n)2/3 n n· g =0. 2 (2a log n)1/2 − n n→∞ n 2 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) Proof : Set Γ := 1 (n/n (cid:12))(cid:12)1/2(a /a )1/2(log(cid:12)(cid:12)n/log n )1/2. The n − k+1 n nk+1 2 2 k+1 triangle inequality yields α (a ) α (a ) H (a ) H (a ) n n· g n n· Γ + n n· − n nk+1· (2a log n)1/2 − n ≤ (2a log n)1/2 n (2n a log a )1/2 (cid:12)(cid:12) n 2 (cid:12)(cid:12) (cid:12)(cid:12) n 2 (cid:12)(cid:12) (cid:12)(cid:12) k+1 nk+1 2 nk+1 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (3.5)(cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) := A +B . n n imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 9 Clearly we have, as k , →∞ n log n max (log n )2/3 Γ 1 k 2 k (log n )2/3 0. n∈Nk 2 k+1 n ≤(cid:18) −snk+1log2nk+1(cid:19) 2 k+1 → Now, by applying (1.4) we have almost surely α (a ) n n limsup · =1. (3.6) (2a log n)1/2 n→∞ n 2 (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) Obviously (3.6) implies that, alm(cid:12)(cid:12)ost surely : (cid:12)(cid:12) lim (log n )2/3 maxA =0. 2 k+1 n n→∞ n∈Nk We now focus on controlling B . Set ρ :=a /a and notice that n k nk nk+1 H (a ) H (a ) P max(log n)2/3 n n· − n nk+1· ǫ (cid:18)n∈Nk 2 (cid:12)(cid:12)(2nk+1ank+1log2ank+1)1/2(cid:12)(cid:12)≥ (cid:19) (cid:12)(cid:12) (cid:12)(cid:12) P max sup (cid:12)(cid:12) (log n )2/3 αn(ank+1ρ(cid:12)t(cid:12))−αn(ank+1t) ǫ . ≤ n∈Nk1≤ρ≤ρk,0≤t≤1 2 k+1 (cid:12) (2ank+1log2nk+1)1/2 (cid:12)≥ ! (cid:12) (cid:12) (cid:12) (cid:12) Now consider the Banach space B [0,1] [0,2] of all real bounded functions × on [0,1] [0,2], endowed with the usual sup norm . We shall now × (cid:0) (cid:1) ||·||[0,1]×[0,2] makeuseofthe powerfulmaximalinequalityofMontgommery-Smith.Forfixed k 1, we apply the just mentioned inequality to the finite sequence (X ) , wi≥th X (t,ρ) := 1 (U ) ρt, t [0,1], ρ [1,ρ ], ρt 1 and X (t,iρ)i∈=Nk0 i [t,ρt] i k i − ∈ ∈ ≤ elsewhere. Hence, by a combination of Theorem 1 and Corollary 3 in [18], we have : α (a ρt) α (a t) P max sup (log n )2/3 n nk+1 − n nk+1 ǫ (cid:18)n∈Nk1≤ρ≤ρk,0≤t≤1 2 k+1 (cid:12) (2ank+1log2nk+1)1/2 (cid:12)≥ (cid:19) α (cid:12) (a ρt) α (a t(cid:12)) 9P sup (log n )2/3 nk(cid:12)+1 nk+1 − nk+1 nk+1(cid:12) ǫ/30 ≤ 2 k+1 (2a log n )1/2 ≥ (cid:18)1≤ρ≤ρk,0≤t≤1 (cid:12) nk+1 2 k+1 (cid:12) (cid:19) (cid:12) ǫ (2n a log n(cid:12) )1/2 18P Π(n a ) W(n (cid:12)a ) k+1 nk+1 2 k(cid:12)+1 ≤ || k+1 nk· − k+1 nk· ||≥ 240 (log (n ))2/3 (cid:18) 2 k+1 (cid:19) W(ρ ) W() ǫ +18P e k· − · . (3.7) (2log n )1/2 ≥ 120(log n )2/3 (cid:18)(cid:12)(cid:12) 2 k+1 (cid:12)(cid:12) 2 k+1 (cid:19) (cid:12)(cid:12) (cid:12)(cid:12) In the last e(cid:12)(cid:12)xpression (which(cid:12)(cid:12)is the combination of usual poissonization tech- niques with the triangularinequality), Π and W denote respectively a centered PoissonprocessandaWienerprocessbasedonthe sameunderlyingprobability space. By the Komlós-Major-Tusnàdyeconstruction (see [15]), W can be con- structedtosatisfy(3.3).Bymakinguseofthesameargumentsasthoseinvoked to obtain(3.4), weconclude thatthe firsttermin(3.7) is sumableink.To con- trolthe secondtermin(3.7), we shallmakeuse ofa wellknown inequality(see, imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012 D. Varron/Clustering rates and Chung limit laws for empirical processes 10 e.g., [20], p. 536), with a:=ρ 1, λ:=(ρ 1)−1/2(log n )−1/6(√2ǫ/120) k− k− 2 k+1 and δ :=1/2, to get W(ρ ) W() ǫ P k· − · (cid:16)(cid:12)(cid:12) 2log2nk+1 (cid:12)(cid:12)≥ 120(log2nk+1)2/3(cid:17) 307(cid:12)(cid:12)20 (cid:12)(cid:12) ǫ2 (cid:12)(cid:12) p(ρ 1)−1/2(((cid:12)l(cid:12)og n )1/6exp (ρ 1)−1(log n )−1/3 . ≤ √2ǫ k− 2 k+1 − 19200 k− 2 k+1 (cid:16) (cid:17) This expression is sumable in k, and hence max B 0 almost surely as k .(cid:3) n∈Nk n → →∞ 3.3. Proof of Theorem 3 Recallthatχ , , bv and liv aredefinedin§2.Themaintooltoachievethe f ∇f S1 S1 proof of Theorem 3 is the following inequality (see Berthet [3]), which sums up different results from Gorn and Lifshits [11], Berthet and Lifshits [4] and Grill [12] (see also de Acosta [1]). Inequality 3.1. For any f BV LIV and δ > 0, there exist γ+ = ∈ S1 ∪ S1 γ+(δ,f)>0 and γ =γ−(δ,f)>0 such that for T sufficiently large : − T2 W T2 2(T2/2) P f (1+δ)χ exp +γ+∇f , ∇f 2 T − ≤ f ≥ − 2 T2 (cid:18) (cid:16) (cid:17)(cid:12)(cid:12) (cid:12)(cid:12) (cid:19) (cid:16) (cid:17) (cid:12)(cid:12) (cid:12)(cid:12) T2 (cid:12)(cid:12)W (cid:12)(cid:12) T2 2(T2/2) P f (1 δ)χ exp γ−∇f . ∇f 2 T − ≤ − f ≤ − 2 − T2 (cid:18) (cid:16) (cid:17)(cid:12)(cid:12) (cid:12)(cid:12) (cid:17) (cid:16) (cid:19) Select f BV(cid:12)(cid:12) LIV(cid:12).(cid:12)We remind the two following properties of (see ∈ S1 (cid:12)(cid:12)∪S1 (cid:12)(cid:12) ∇f [16]), namely limsup L−1 (L) < and liminf L−2/3 (L) > 0. L→∞ ∇f ∞ L→∞ ∇f We shall first show that, almost surely : α (a ) n n liminf (log n) · f χ . n→∞ ∇f 2 (2anlog2n)1/2 − ≥ f (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) (cid:12)(cid:12) Letus fix ǫ>0.We startby apply(cid:12)i(cid:12)ngpoissonizationte(cid:12)c(cid:12)hniques incombination with the Komlós-Major-Tusnàdyapproximation. α (a ) P (log n) n n· f χ (1 2ǫ) ∇f 2 (2a log n)1/2 − ≤ f − n 2 (cid:16) (cid:12)(cid:12) (cid:12)(cid:12) (cid:17) (cid:12)(cid:12) W(na ) (cid:12)(cid:12) 2P (log n(cid:12))(cid:12) n· (cid:12)(cid:12)f χ (1 ǫ) ≤ ∇f 2 (2na log n)1/2 − ≤ f − n 2 (cid:16) (cid:12)(cid:12) (cid:12)(cid:12) (cid:17) (cid:12)(cid:12) χ ǫ((cid:12)2(cid:12)na log n)1/2 +2P W(na(cid:12)(cid:12)) Π(na ) f (cid:12)(cid:12) n 2 . n n · − · ≥ (log n) (cid:16)(cid:12)(cid:12) (cid:12)(cid:12) ∇f 2 (cid:17) These two terms are(cid:12)(cid:12)(cid:12)(cid:12)sumable aloneg the su(cid:12)(cid:12)(cid:12)(cid:12)bsequence n , the second term being k controlled by the Komlós-Major-Tusnàdy approximation while the first one is controlled by Inequality 3.1. Now the control between n and n follows the k k+1 imsart-generic ver. 2007/12/10 file: Vitrec.tex date: January 27, 2012