FUNCTIONAL LAWS OF THE ITERATED LOGARITHM FOR THE INCREMENTS OF THE COMPOUND EMPIRICAL PROCESS 1 2. Abstract

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1 FUNCTIONAL LAWS OF THE ITERATED LOGARITHM FOR THE INCREMENTS OF THE COMPOUND EMPIRICAL PROCESS 1 2 Myriam MAUMY Received: Abstract Let {Y i : i 1} be a sequence of i.i.d. random variables and let {U i : i 1} be a sequence of independent and uniformly distributed random variables on, 1). The two sequences are assumed to be independent. Let {α n,c t) : t 1} be the compound empirical process generated by the first n observations from these two sequences of random variables. Let < h n < 1 be a sequence of constants such that h n as n. We investigate the strong limiting behavior as n of the increment function {α n,c x + h n u) α n,c x) : u 1}, where x 1 h n. Under suitable regularity assumptions imposed upon h n, we prove functional laws of the iterated logarithm for this increment function. 1. Notation Let U 1, U 2,..., be a sequence of independent uniform, 1) random variables rv and for each integer n 1, let U n x) = 1 n 1 {Ui x}, x 1, n i=1 be the right-continuous empirical distribution function, based on the first n of these uniform, 1) random variables. Here, 1 A denotes the indicator function of the event A. The uniform empirical process will be denoted for each integer n 1, by α n x) = n U n x) x, x 1. 1 AMS classification: 6F17 2 Keywords: Laws of the iterated logarithm, compound empirical process.

2 MAUMY 2 For any < h < 1 and integer n 1 consider the increment function 1.1) ξ n h, x; u) = α n x + hu) α n x), for u 1 and x 1. Let Y = Y 1, Y 2,..., be a sequence of independent and identically distributed iid random variables. Moreover, we suppose that EY = 1 and that < σ 2 = VarY <. The sequences {U i : i 1} and {Y i : i 1} are assumed to be independent. For each integer n 1, let U n,c x) = 1 n Y i 1 {Ui x}, x 1, n i=1 be the right-continuous compound empirical distribution function, based on the first n of these random variables. Here, the subscript c is used with the meaning of compound. The compound empirical process will be written for each integer n 1 by 1.2) α n,c x) = n U n,c x) x, x 1. For any < h < 1 and integer n 1 consider the increment function 1.3) ξ n,c h, x; u) = α n,c x + hu) α n,c x), for u 1 and x 1. A sequence of constants h n ) n 1, will be said to satisfy the Csörgő-Révész-Stute see, e.g., Csörgő-Révész 4, Stute 15) conditions CRS if the following hold: S.1) < h n < 1 for n 1, h n and nh n as n ; S.2) log1/h n )/ log log n as n ; S.3) nh n / log n as n. At times, we will make use of the condition S.4) nh n / log log n as n. The Strassen set S see, e.g., Strassen 14) consists of all absolutely continuous functions f on, 1 such that 1 2 f) = and fs)) ds 1, where f denotes the Lebesgue derivative of f. Let B, 1) be the set of all bounded functions on, 1. The sup-norm of a function f B, 1) is denoted by f = sup fs). s 1 For any ε > and C B, 1), we denote by C ε the set of all functions f B, 1) such that there exists a g C with f g < ε.

3 3 MAUMY 2. Strassen-type functional laws of the iterated logarithm for the increments of compound empirical processes 2.1. Introduction The tail empirical process see, e.g., Mason 9) is defined for each integer n 1 by h 1/2 n α n hn x ), x 1, where α n is the uniform empirical process, and h n is a sequence of positive constants such that h n and n h n as n. In order to motivate the results of this section, we cite the functional laws of iterated logarithm obtained by Mason 1988) for the tail empirical process based on α n. This result is in stated in Theorem 2.1, below. THEOREM 2.1. Under S.1) and S.4), the sequence of functions 2.1) { αn h n x) 2hn log log n : x 1 } are almost surely relatively compact in B, 1) endowed with the topology of uniform convergence on, 1. Moreover, the set of limit points is equal to the Strassen set S. Proof. See Mason 9. Next, we state in Theorem 2.2 below the functional laws of iterated logarithm due to Deheuvels and Mason 1992) for the increment process ξ n, as defined in 1.1). THEOREM 2.2. Under the CRS conditions S ), for any ε >, there exists almost surely a finite n ε such that, for all n n ε, we have 2.2) { } ξ n h n, x;.) 2hn log1/h n ) : x 1 h n S ε. Moreover, for any f S and each ε >, there exists almost surely a finite n ε,f for all n n ε,f, there exists an x, with x = x n,ε,f 1 h n, and such that 2.3) ξ n h n, x;.) 2hn log1/h n ) f < ε. such that, Proof. See Deheuvels and Mason 5.

4 MAUMY Main result We now state the versions of functional laws of iterated logarithm which hold for the increment process ξ n,c of the compound empirical process as defined in 1.3). Our main result is as follows. THEOREM 2.3. Suppose that the sequence {Y i : i 1} of i.i.d.r.v is such that : EY i = 1, VarY i = σ 2 < and Y i 1 M, for i 1. Under the CRS conditions S ), for any ε > there exists almost surely a finite n ε such that, for all n n ε, we have 2.4) { ξ n,c h n, x;.) 2σ2 + 1)h n log1/h n ) : x 1 h n } S ε. Moreover, for any f S and each ε >, there exists almost surely a finite n ε,f for all n n ε,f, there exists an x, with x = x n,ε,f 1 h n, and such that ξ n,c h n, x;.) 2.5) 2σ2 + 1)h n log1/h n ) f < ε. 3. Preliminary results on compound processes 3.1. Results on compound Poisson process A compound Poisson process {Π c t) : t } with rate λ is defined, by Πt) 3.1) Π c t) = Y i, t, i=1 such that, where {Πt) : t } denotes a Poisson process with rate λ >, and {Y i : i 1} is a sequence of independent and identically distributed random variables with common distribution function F y) = PY y, where Y = Y 1. The Poisson process {Πt) : t } and the sequence {Y i : i 1} are assumed to be independent. Recall that the subscript c in Π c is used with the meaning of compound. Remark 3.1. In 3.1), we set Y i =. i=1 THEOREM 3.1. The compound Poisson process {Π c t) : t } with rate λ based upon {Y i : i 1} has stationary independent increments, and characteristic function defined, for any t, by ϕ Πct)u) = exp λt 1 ϕ Y u)), where ϕ Y u) is the characteristic function of Y. Subject on EY 2 <, Π c t) has finite second moments given, for t, by EΠ c t) = µλt, VarΠ c t) = σ 2 + µ 2 )λt, where µ and σ 2 are the mean and variance, respectively, of Y.

5 5 MAUMY Proof. See, for example, Karlin and Taylor 7. THEOREM 3.2. The centered compound Poisson process {Π c t) µλt : t } is a martingale with respect to the σ-algebras A s = σ{π c u) : u s}. Proof. See, e.g., Maumy A useful Poisson Approximation Theorem Our next result relates compound empirical measures to compound Poisson processes. This uses the fact that compound Poisson processes are compound empirical measures. It goes as follows. Consider a separable metric space E with Borel algebra of subsets B. Denote by µ a bounded Borel positive measure on E. Next, consider an independent and identically distributed sequence of E-valued random variables ξ 1, ξ 2,..., with probability distribution Pξ B) = µ B) = µb) µe), for B B. Consider also an independent and identically distributed sequence of random variables Y = Y 1, Y 2,..., whose follows the common distribution function F and such that EY = 1. Moreover the sequence of random variables {Y i : i 1} is independent of the sequence of E-valued random variables {ξ i : i 1}. Define now a random variable Nλ), independent of ξ 1, ξ 2,..., independent of Y = Y 1, Y 2,..., and following a Poisson distribution with mean λ >, i.e. PNλ) = k = λk exp λ) for k N. k! The Poisson process on E with mean measure ν = λµ is defined as the random point measure on E given by Π = Nλ) i= The compound Poisson process on E, R) with mean measure ν c = λµ EY = λµ is defined as the marked point measure on E, R) given by Π c = Nλ) i= δ ξi. Y i δ ξi, where we use the convention that.) =. Now, the most important feature of the random measure Π c is that, for any collection A 1, A 2,... of disjoint Borel subsets of E, R), the random variables Π c A 1 ), Π c A 2 ),..., are independent. Moreover, for each A B, R), the random variable Π c A) follows a compound Poisson distribution with mean ν c A). It is not difficult to check that, if Π 1,c and Π 2,c are two independent compound Poisson processes with mean measures ν 1,c and ν 2,c, then the superposition Π 1,c + Π 2,c of Π 1,c and Π 2,c is a compound Poisson process with mean ν 1,c + ν 2,c see, for example 7 p ).

6 MAUMY 6 Given this fact, we may get rid of our previous assumption that the mean measures of the compound Poisson processes are bounded by defining a general compound Poisson process with mean measure ν c = ν i,c. i= This process is obtained by superposing a countable sequence of independent compound Poisson processes, with mean measures {ν i,c : < i < }. Of particular interest is the case where ν c is the compound measure on R and ν i,c is the uniform measure on i, i+1 weighted by Y i. In this case, one obtains the compound Poisson process on the line. If we denote this process by Π c,, it is convenient to define Π c, on R + by its cumulated right-continuous distribution function ) Π c t) = Π c,, t for t 1. Given these facts, the following Theorem is straightforward. We denote by d = equality in distribution. THEOREM 3.3. We have the following distribution identity. For any λ >, Proof. Omitted. {nu n,c t) : t 1} d = {Π c λt) : t 1 Nλ) = n}. Since the choice of λ > in Theorem 3.3 is arbitrary, it is convenient to choose λ = n. By so doing, we have the equality of expectations, with Π n,c t) = Π c nt), EnU n,c t) = EΠ c nt) = EΠ n,c t) = nt, for t 1. Below, we denote by B h the Borel σ-algebra of subsets induced on RC, h endowed with the topology of uniform convergence). THEOREM 3.4. For each < h < 1, there exists a constant < C h < such that, for all n 1 and all B B h, P {nu n,c t) : t h} B C h P {Π n,c t) : t h} B. Proof. Consider first a λ > and a random variable Z following a Poisson P oλ) distribution, i.e., such that It is easily checked that PZ = k = λk exp λ) for k N. k! λ λ 3.2) Mλ) = max PZ = k = PZ = λ ) = exp λ), k N λ! where u u < u + 1 denotes the integer part of u. Some routine calculus making use of the Stirling formula that is n! = 1 + o1))n/e) n 2πn as n ) shows that, as λ, 3.3) Mλ) = 1 + o1) 2πλ.

7 7 MAUMY Now, under the assumptions of Theorem 3.4, set R = Nnh) and R = Nn) Nnh). Observe that R and R are independent and follow Poisson distributions, with respective parameters P onh) and P on1 h)). Since R+R = Nn) follows also a Poisson distribution, with parameter P on), it follows from 3.2)-3.3) that, as n, 3.4) sup k N PR = k PR + R = n Mn1 h)) = ) = Mn) ) / 1 ) 1 + o1) + o1) 1. 2πn1 h) 2πn 1 h It follows from 3.4) that we may define a finite constant C h by setting { } ) Mn1 h)) 1 3.5) C h = sup,. n 1 Mn) 1 h Introduce now the events E 1 = { {nu n,c t) : t h} B } and E 2 = { {Π n,c t) : t h} B }. Observe, via Theorem 3.3 and 3.4)-3.5), that, uniformly over B B h, P E 1 = P E 2 R + R = n n PR = k = P E 2 {R = n k} PR + R = n which was to be proved. k= PR = k P E 2 sup k N PR + R = n C h P E 2, 3.3. Results on the modulus of continuity of compound processes Let h be a real number such that < h < 1 and let C be an interval s, t in, 1, that C = t s, and that XC) = Xt) Xs) denotes the increment. Introduce now the modulus of continuity ω X of the X process defined by ω X h) = sup XC) = sup Xt) Xs) = sup C h t s h sup u h t 1 u Xt + u) Xt). INEQUALITY 3.1. Let h, δ be some real numbers such that < h δ 1. Suppose 2 that the i.i.d.r.v. Y = Y 1,, Y n are such that Y i EY i M for i = 1,, n and VarY = σ 2 <. Then, for each n 1 and β >, we have P ω Πn,c h) β nh 2 )) hδ exp β2 1 δ) σ ψ Mβ σ 2 + 1). nh Proof. See, e.g., Maumy 11. INEQUALITY 3.2. Let h, δ be some real number such that < h δ 1. Suppose 2 that the i.i.d.r.v. Y = Y 1,, Y n are such that Y i EY i M for i = 1,, n and VarY = σ 2 <. Then, for each n 1 and β >, we have 3.6) P ω αn,c h) β h 2 )) hδ C 3 h exp β2 1 δ) 3 2 σ ψ Mβ σ 2 + 1). nh Proof. Omitted.

8 MAUMY The compound empirical process is a martingale Finally, we shall show that behind the compound empirical process there is a martingale structure. In order to prove this result, we shall define for each n 1 and t 1 the σ-algebras F n,c t) = σ Y i, 1 {Ui s} : 1 i n, s t. Introduce for each n 1 and t 1, 3.7) M n,c t) = n t U n,c t) 1 U n,c s) ds. 1 s LEMMA 3.1. For each n 1, M n,c t), F n,c t) ), t 1, is a compensator. Proof. Consider the counting process N i t) = 1 {Ui t}, t, 1, i {1,..., n} see Jacod, 6). For each i such that 1 i n, denote by M i t) = N i t) t and let M c it) be its compound version defined by M c it) = Y i N i t) t for each i such that 1 i n. Note that, for all s < t, 1 {Ui s} ds, t 1, 1 s Y i 1 {Ui s} 1 s ds = Y im i t), t 1, EM c it) F n,c s) = EY i M i t) F n,c s) = Y i EM i t) F n,c s). We recall from Jacod 6, that M i t) is a compensator with respect to the family of σ-algebras F n t) = σ1 {Ui s} : 1 i n, s t. Since the sequence {Y i : i 1} is independent of the independent and uniformly distributed on, 1) random variables U 1,..., U n, we have, for all s < t 1, EM c it) F n,c s) = Y i EM i t) F n,c s) = Y i EM i t) F n s) = Y i M i s) = M c is), be which we conclude that M c it), F n,c t) ), t 1 is a compensator. Moreover, since a sum of compensators is also a compensator and since 3.7) holds, M n,c t) = n M c it), is a compensator on, 1 with respect to {F n,c t)}. This achieves the proof. i=1 LEMMA 3.2. We have for each n 1, ) ) αn,c t) Un,c t) t n = n = 1 t 1 t Proof. We have I n = 1 n = = t t t t t 1 1 s dm n,cs), t s dm 1 n,cs) = 1 s du 1 U n,c s) n,cs) + d1 s) 1 s)1 s) 1 t ) 1 1 s du n,cs) 1 U n,c s)) d 1 s t s du Un,c s) t 1 n,cs) + 1 s 1 s d1 U n,cs)) = 1 ) αn,c t). n 1 t = U n,ct) t 1 t t

9 9 MAUMY For convenience, we set for each n 1 and t 1 ) Un,c t) t M n,c t) = n = ) αn,c t) n. 1 t 1 t Lemma 3.3 establishes that M n,c t) is a martingale with respect to {F n,c t)}. LEMMA 3.3. M n,c t), F n,c t)), t 1, is a martingale. Proof. We show that M n,c t) is a martingale with respect to F n,c t) by using the fact the integral of a predictable process with respect to a martingale is again a martingale, provided that the appropriate moments exist see Shorack and Wellner 13, p.261). 4. Proof of Theorem 2.3 In the remainder of this section, we present a proof of Theorem 2.3. Throughout and unless otherwise specified, we assume that the CRS conditions S ) are satisfied. For < h < 1, let 4.1) L n,c h, x, u) = Π n,cx + hu) Π n,c x) nhu n, for u 1 and x 1. LEMMA 4.1. There exists a constant C hn with < C hn <, and such that the following property holds. For each choice of {x 1,, x m } {kh n : k h 1 n 1} with < mh n 1 2 and Borel subsets B 1,..., B m of B, 1) endowed with the topology of uniform convergence on, 1, if A 1 = {ξ n,c h n, x i ;.) B i, i = 1,..., m} and then A 2 = {L n,c h n, x i ;.) B i, i = 1,..., m}, PA 1 C hn PA 2. Proof. The proof of Lemma 4.1 is practically the same as the proof of Theorem 3.4 after a change of scale. Therefore, we omit details. The next lemmaknown as Schilder s Theorem see, e.g., Schilder 12) gives a large deviation result which will be instrumental for our needs. For any f B, 1), set 1 ) 2ds, when f is absolutely continuous on, 1 fs) Jf) = with Lebesgue derivative f,, otherwise, and, for any B B, 1), define JB) = inf f B Jf).

10 MAUMY 1 LEMMA 4.2. Let {W x) : x } be a standard Wiener process and, for any γ >, set W γ x) = 2 1/2 γ 1 W γx), for x 1. Then the following properties hold: i) For each closed subset F of B, 1) endowed with the topology of uniform convergence on, 1, 4.2) lim sup γ 1 log PW γ F ) JF ). γ ii) For each open subset G of B, 1) endowed with the topology of uniform convergence on, 1, 4.3) lim inf γ γ 1 log PW γ G) JG). Proof. For the proof of Lemma 4.2, we refer to 16. The following lemma establishes the second part ie 2.5) of Theorem 2.3 for ξ n,c. LEMMA 4.3. Under the CRS conditions, for every f S and ε >, there exists almost surely a finite n ε,f such that, for all n n ε,f, there exists a x = x n,ε,f, 1 h n such that: ξ n,c h n, x;.) 2σ2 + 1)h n log1/h n ) f < ε holds. Proof. Set N ε f) = {g B, 1) : f g < ε}. By Lemma 4.1 applied with x i = ih n, i = 1,, m n = 1/2h n and B i = B, 1) N ε f), for i = 1,, m n, we obtain 4.4) P n = P mn C hn P ) ξ n,c h n, x i ;.) 2σ2 + 1)h n log1/h n ) N εf) i=1 mn i=1 L n,c h n, x i ;.) 2σ2 + 1)h n log1/h n ) N εf) L n,c h n, ;.) = C hn 1 P 2σ2 + 1)h n log1/h n ) N εf) = C hn 1 P1,n ) mn. ) ) mn In a second step, we evaluate P 1,n. For this, we apply the strong invariance principles of Komlós, Major and Tusnády 8 to the compound Poisson process. These enable us to construct on the same probability space a compound Poisson process {Π c t) : t }, and a standard Wiener process {W t) : t }. We have the following theorem.

11 11 MAUMY THEOREM 4.1. We can define a Wiener process {W t) : t } such that : if EexptY ) < in a neighbourhhod of, then / 4.5) P sup Π c t) µt) σ2 + µ 2 W t) C 1 log T + z C 2 exp C 3 z), t T for all z >, where C 1, C 2 and C 3 are positive constants. Proof. See, e.g., Csörgő, Deheuvels, Horváth 3. By 4.1), 4.4) and 4.5), we see that L n,c h n, ; ) P 1,n = P 2σ2 + 1)h n log1/h n ) N εf) Π c nh n ) nh n = P 2σ2 + 1)nh n log1/h n ) N εf), with Π c ) =. Then, we have 4.6) W nh n ) P 1,n P 2nhn log1/h n ) N ε/2f) P Π c nh n u) nh n u) / σ W nh n u) sup u 1 2nhn log1/h n ) ε/2 = P 2,n P ε/2 3,n. Notice that W nh n ) 2nhn log1/h n ) d = 2 1/2 log1/h n ) ) 1 W log1/hn ) ), under the S.2 3) conditions, so that where P 2,n = P W log1/hn) ) N ε/2 f), W γ x) = 2 1/2 γ 1 W γx), for x 1. Since f S, we have obviously JN ε/2 f)) < 1. Thus, by 4.3), it follows that for any ρ JN ε/2 f)), 1), we have for all n sufficiently large 4.7) P 2,n exp ρ log1/h n )) = h ρ n. Next, observe that S.1) implies that lognh n ) 2nhn log1/h n ) as n.

12 MAUMY 12 This, when combined with 4.5), 4.6) and S.3), implies that, for large n, P ε/2 Πc 3,n P sup x) x) / σ W x) x nh n 4.8) C 1 lognh n ) + ε 4 2nhn log 1/h n ) ε ) C 2 exp C 3 2nhn log 1/h n ) 4) 1 2 exp ρ log 1/h n)) = 1 2 hρ n. By 4.6), 4.7) and 4.8), we have ultimately as n, P 1,n P 2,n P ε/2 3,n h ρ n 1 2 hρ n P 1,n 1 2 hρ n, which, when combined with 4.4), yields for all large n the inequalities P n C hn exp 1 ) 2 m nh ρ n, P n C hn exp 1 ) ) 1 h ρ n, 2 4h n P n C hn exp 1 ) 4.9) 8 hρ 1 n. Since ρ 1 <, by S.2), the right hand side of 4.9) is ultimately less than or equal to C hn exp 18 ) log n)r for an arbitrary r > 1. It follows that n P n <, which by the Borel-Cantelli lemma implies 2.5), as sought. We now turn to the proof of 2.4). Fix any γ > and ε >. Set ν k = 1 + γ) k, for k 1, and let b n = 2σ 2 + 1)h n log1/h n ). Next, consider the events { C k ε, γ) = n/ ) 1/2 b 1 ξ n,c h νk+1, x; ) S ε for some x 1 h νk+1 and ν k < n }, and D k ε, γ) = { } b 1 ξ νk+1,c hνk+1, x;.) S ε for some x 1 h νk+1.

13 13 MAUMY LEMMA 4.4. For every ε > and γ >, there exists a K = K ε,γ such that k K implies that 4.1) P C k ε, γ) C hn P D k ε/2, γ). Proof. Let r i, i 1 be a denumeration of the rationals in, 1 h νk+1, and introduce the events for i 1 and ν k < n, { } E k,i,n ε) = n/ ) 1/2 b 1 ξ n,c h νk+1, r i ; ) S ε, E k,n ε) = i 1 E k,i,n ε) and F k,i,n ε) = { } b 1 ξνk+1,ch νk+1, r i ; ) n/ ) 1/2 ξ n,c h νk+1, r i ; ) < ε. Observe that for any ε 1 > and ε 2 >, the sequences of events {E k,i,n ε 1 ) : i 1} and {F k,i,n ε 2 ) : i 1} are independent. Denote by A the complement of the event A. We have i 1 q 1 P C k ε, γ) = P E k,i,q ε) E k,j,q ε) E r,q ε), and hence 4.11) inf ν k <n q=ν k +1 i=1 j=1 inf P F k,m,nε/2) P C k ε, γ) m 1 P E k,i,q ε) F k,i,q ε/2) q=ν k +1 i=1 i 1 j=1 ) P E k,i,νk+1 ε/2 = P D k ε/2, γ). i=1 Next, we see that, for any < h 1, β > and n 1, we have 2 P n α n,c u) β P sup u h r=ν k +1 E k,j,q ε) q 1 r=ν k +1 sup n U n,c u) u /1 u) β u h E r,q ε) which by the fact nu n,c u) u)/1 u) is a martingale in u see Preliminary results), is P n αn,c u) β E n 2 U n,c h) h 2 / β1 h)) 2 σ 2 + 1)nh/β 2. sup u h From this last inequality, we obtain that, for all ν k < n, m 1 and all k large enough, P F k,m,n ε/2) = P ν 1/2 k+1 sup n) U νk+1 n,cu) u ε/2)b νk+1 u h νk+1 / ) 4σ 2 + 1) ν k )h νk+1 ε 2 b 2 Using this bound in 4.11) yields 4.1). 2σ 2 + 1) / ε 2 log1/h νk+1 ) ) as k.,

14 MAUMY 14 LEMMA 4.5. We have lim lim sup max sup γ k ν k <n x 1 h νk ) n/ ) 1/2 b 1 b 1 n ξn,c h νk+1, x; ) ) = a.s. Proof. The left hand side 4.12) of is less than or equal to bn 4.13) lim lim sup max b 1 ν γ k ν k <n /n ) ) 1/2 lim sup b 1 n ω αn,c h n ), k+1 n where ω αn,c h n ) is defined, for < h n < 1 and n 1, by ω αn,c h n ) = sup ξn,c h n, x; ). x 1 h n Making use of the Theorem 2.1 in Maumy 1, we have 4.14) lim n b 1 n ω αn,c h n ) = 1 a.s. Moreover, the condition S.1) implies that nh n and h n as n, so that b n = 2σ2 + 1)h n log1/h n ) is ultimately nonincreasing and such that, for ν k < n, b n /b νk+1 1 b νk /b νk o1) ) /ν k ) 1 γ as k. This, in combination with 4.13) and 4.14), readily yields 4.12). LEMMA 4.6. We have 4.15) lim γ lim sup k max sup b 1 n ν k <n x 1 h νk+1 Proof. From h n we get that for ν k < n, ξn,c h νk+1, x; ) ξ n,c h n, x; ) ) = a.s. h n h νk+1 = h n 1 hνk+1 /h n ) hn 1 hνk+1 /h νk ). From nh n we obtain that, for all k large enough, h νk+1 /h νk ν k / 1 γ. Therefore, for ν k < n and for all k large enough, h n h νk+1 h n 1 hνk+1 /h νk ) γhn. Hence the left hand side of 4.15) is less than or equal to ) 4.16) lim lim n ω αn,c γh n ). γ n b 1 By 4.14) and 4.16) it is easy to conclude 4.15).

15 15 MAUMY In view of Lemmas , we will show in the sequel that the proof of 2.4) boils down to showing that, for every ε > and γ >, 4.17) P D k ε, γ) <. k=1 Toward this end, we will make use of the following inequality. LEMMA 4.7. For all < θ < 1, ε > and n 1, we have P b 1 n ξ n,c h n, x; ) S ε for some x 1 h n θh n ) 1 P b 1 n ξ n,c h n, ; ) S ε/2 + P b 1 n ω αn,c θh n ) > ε/4 Proof. For θih n x θi + 1)h n and i, 1,..., h 1 n 1)/θ 1 = µ n 1, or for µ n h n x 1 h n and i = µ n, we have uniformly over u 1, ξ n,c h n, x; u) ξ n,c h n, iθh n ; u) α n,c x + uh n ) α n,c iθh n + uh n ) The remainder of the proof is obvious. + αn,c x) α n,c iθh n ) 2ω αn,c θh n ). We are now prepared to show that 4.17) holds for all ε > and γ >. We have use Lemma 4.7 to obtain that, for any θ > sufficiently small, P D k ε, γ) ) 1 θh νk+1 P b 1 ξ νk+1,ch νk+1, ;.) S ε/2 + P b 1 ω ανk+1,cθh νk+1 ) ε/4 Q 1,k + Q 2,k. Next, we use Lemma 4.1 to obtain the inequality ) θhνk+1 Q1,k = P b 1 ξ νk+1 h νk+1, ;.) S ε/2 C hn P b 1 Lh νk+1, ;.) S ε/2 = C hn Q 3,k. We now follow the arguments of the proof of Lemma 4.3 to obtain the following analogue of 4.6), with P ε/2 3,n being as in 4.8) ) 4.18) Q 3,k P W log1/hνk+1 ) S ε/4 + P ε/4 3, = Q 4,k + Q 5,k. Since B, 1) S ε/4 is closed with respect to the topology of uniform convergence) and obviously satisfies JB, 1) S ε/4 ) > 1, it follows from 4.2) that, for any ρ 1, JB, 1) S ε/4 )), we have, for all k sufficiently large, 4.19) Q 4,k exp ρ log1/h νk+1 ) ) = h ρ.

16 MAUMY 16 Moreover, the same arguments as used for 4.7) and 4.8) show that, for all k sufficiently large, 4.2) Q 5,k h ρ. Thus, combining 4.18), 4.19) and 4.2), we obtain, for all large k, 4.21) Q 1,k C hn θhνk+1 ) 1 Q3,k 2C h n θ h ρ 1 = 2C h n exp ρ 1 ) )) 1 log. θ h νk+1 Since S.2) implies that log1/h νk+1 )/ log k as k, we see from 4.21) that Q 1,k 1/k 2 for all large k, so that 4.22) Q 1,k <. k=1 We use 3.6) taken with δ = 1/2, h = θh νk+1, n =, β = obtain 4.23) Q 2,k 16 C θhνk+1 θh νk+1 ) 1 exp ε2 log1/h νk+1 ) ψ 128θ ε 4 θ 2σ 2 + 1) log1/h νk+1 ) to Mε 2 log1/hνk+1 ) 4θ σ 2 + 1) h νk+1 Since S.3) implies that nh n / log 1/h n ), and ψx) 1 as x, the right hand side of 4.23) is less than or equal to ) )) ε 16 C θhνk+1 θ exp 256θ 1 log. h νk+1 Thus, the same argument as used for 4.22) shows that, for any < θ < ε/128, we have )). 4.24) Q 2,k <. k=1 Combining 4.22) and 4.24), we see that 4.17) holds as sought. By the Borel-Cantelli lemma and Lemmas , it follows that for any ε >, whenever γ > is sufficiently small, the event ν k <n { b 1 n ξ n,c h n, x; ) : x 1 h νk+1 } S ε holds finitely often with probability 1. The following lemma shows that the same is true for the event { } b 1 n ξ n,c h n, x; ) : 1 h νk+1 < x 1 h n S ε, ν k <n this completing the proof of 2.4).

17 17 MAUMY LEMMA 4.8. We have 4.25) lim γ lim sup k sup sup b 1 n ν k <n 1 h νk+1 x 1 h n ξn,c h n, x; ) ξ n,c h n, 1 h νk+1 ; ) ) = a.s. Proof. By the same arguments as used in the proof of Lemma 4.6, we see that the right hand side of 4.25) is less than or equal to ) lim 2 lim b 1 n ω αn,c γh n ), γ n which equals a.s. by 4.14) and 4.16). References 1 P. Billingsley. Convergence of Probability Measures. Wiley, Y. S. Chow and H. Teicher. Probability Theory. Springer-Verlag, M. Csörgő, P. Deheuvels, and L. Horváth. An approximation of stopped sums with apllications in queueing theory. Adv. Appl. Prob., 19:674 69, M. Csörgő and P. Révész. Strong Approximations in Probability and Statistics. Academic Press, P. Deheuvels and D. M. Mason. Functional laws of the iterated logarithm for the increments of empirical and quantile processes. Ann. Probab., 2: , J. Jacod. Multivariate point processes : predictable projection, radon-nykodim derivatives, representation of martingales. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 31: , S. Karlin and H. M. Taylor. A Second Course in Stochastic Processes. Academic Press, J. Komlós, P. Major, and G. Tusnády. An approximation of partial sums of independent rv s and the sample d.f. i. Z. Wahrscheinlichkeitstheorie verw. Gebiete, 32: , D. M. Mason. A strong invariance theorem for the tail empirical process. Ann. Inst. H. Poincarré, 24:491 56, M. Maumy. Le comportement des oscillations du processus empirique composé. C. R. Acad. Sci. Paris, Ser.I, 333: , M. Maumy. Sur les oscillations du processus de poisson composé. C. R. Acad. Sci. Paris, Ser.I, 334:75 78, 22.

18 MAUMY M. Schilder. Some asymptotic formulas for wiener integrals. Trans. Amer. Math., 134: , G. R. Shorack and J. A. Wellner. Empirical Processes with Applications to Statistics. Wiley, V. Strassen. An invariance principle for the Law of the Iterated Logarithm. Z. Wahrscheinlichkeitstheorie und verw. Gebiete, 3: , W. Stute. The oscillation behaviour of empirical processes. Ann. Probab., 1:86 17, A. D. Ventsel. Rough limit theorems on large deviations for markov processses. Theory Probab. Appl., 21: , , L.S.T.A., Université Paris VI Boîte Courrier 158, 8 e étage, Plateau A 175, rue du Chevaleret, 7513 Paris, France address : maumy@ccr.jussieu.fr