Optimal Hölderian functional central limit theorems for uniform empirical and quantile

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Optimal Hölderian functional central limit theorems for uniform empirical and quantile processes Djamel Hamadouche Laboratoire de Mathématiques Faculté des Sciences Université M. Mammeri Tizi-Ouzou Algérie Charles Suquet Laboratoire P. Painlevé (UMR 8524 CNRS) U.F.R. de Mathématiques Bâtiment M2 Université des Sciences et Technologies de Lille F-59655 Villeneuve d'Ascq Cedex France Abstract Let Ho? be the Hölder space of functions x : [0, 1] ? R such that |x(t + h)? x(t)| = o(?(h)) uniformly in t ? [0, 1], where ?(h) = h?L(1/h) with 0 < ? < 1 and L is normalized slowly varying. Denote by ?pgn the polygonal smoothing of the uniform empirical process. We prove that ?pgn converges weakly in H o ? to the Brownian bridge B if and only if h1/4 = o(?(h)). We also prove that the polygonal smoothing ?pgn of the uniform quantile process converges weakly in Ho? to B if and only if h1/2 ln(1/h) = o(?(h)). Keywords : Brownian bridge, empirical process, Hölder space, quantile process random polygonal line, spacings. MSC 2000 : 60B12, 60F17, 62G30.

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Optimal Hölderian functional central limit theorems for uniform empirical and quantile processes Djamel Hamadouche Laboratoire de Mathématiques Faculté des Sciences Université M. Mammeri Tizi-Ouzou Algérie Charles Suquet Laboratoire P. Painlevé (UMR 8524 CNRS) U.F.R. de Mathématiques Bâtiment M2 Université des Sciences et Technologies de Lille F-59655 Villeneuve d’Ascq Cedex France

Abstract Let H ρo be the Hölder space of functions x : [0 , 1] → R such that | x ( t + h ) − x ( t ) | = o ( ρ ( h )) uniformly in t ∈ [0 , 1] , where ρ ( h ) = h α L (1 /h ) with 0 < α < 1 and L is normalized slowly varying. Denote by ξ p n g the polygonal smoothing of the uniform empirical process. We prove that ξ pg converges weakly in H oρ to the Brownian bridge B if and only n if h 1 / 4 = o ( ρ ( h )) . We also prove that the polygonal smoothing χ p n g of the uniform quantile process converges weakly in H oρ to B if and only if h 1 / 2 ln(1 /h ) = o ( ρ ( h )) . Keywords : Brownian bridge, empirical process, Hölder space, quantile process random polygonal line, spacings. MSC 2000 : 60B12, 60F17, 62G30. 1 Introduction Let U 1 , . . . , U n , . . . be independent and [0 , 1] uniformly distributed random vari-ables. Let F n ( t ) be the empirical distribution function based on U 1 , . . . , U n and G n ( t ) be the empirical quantile function. It is well known that the empirical and quantile processes ξ n ( t ) := √ n  F n ( t ) − t  , χ n ( t ) := √ n  G n ( t ) − t  , t ∈ [0 , 1] , 1

both converge in law to the Brownian bridge B . The usual topological setting for this convergence is the Skorohod’s space D (0 , 1) . Changing the topological framework changes the set of continuous functionals of the paths and hence mod-iﬁes the scope of the convergence in law of ξ n and χ n . For instance, Dudley [8] proves that the convergence of ξ n to B holds with respect to the p -variation norm for p ∈ (0 , 2) . In another direction, Morel and Suquet [14] give a neces-sary and suﬃcient condition for the convergence in law in L 2 (0 , 1) of ξ n to a Gaussian process when the U i ’s are associated. Under the same positive depen-dence assumption, they also investigate the convergence in law of ξ n in some Besov spaces. Smoothing ξ n and χ n allows us to look for stronger topologies. Denote by F n pg the polygonal cumulative empirical distribution function (which interpo-lates linearly F n between its consecutive jumps). Denote by G n pg the polygonal uniform sample quantile function (precise deﬁnitions are given in subsection 2.1). Then deﬁne the processes ξ n pg ( t ) := √ n  F n pg ( t ) − t  , χ n pg ( t ) := √ n  G n pg ( t ) − t  , t ∈ [0 , 1] . It is well known that ξ p n g and χ n pg converge in law to B in the space C[0 , 1] of continuous functions endowed with the supremum norm. Because the Hölder spaces are topologically embedded in C[0 , 1] , they support more continuous functionals than C[0 , 1] . From this point of view, the alternative framework of Hölder spaces gives functional limit theorems of a wider scope than C[0 , 1] . This choice may be relevant when the paths of ξ p n g and χ p n g and of the limit process B share some α -Hölder regularity. Due to the well known regularity of Brownian paths, this requirement is clearly satisﬁed with any α < 1 / 2 and it seems rather natural to ask for some Hölderian convergence in law of ξ n pg and χ n pg . Considering the Hölder spaces H oα of functions x : [0 , 1] → R such that | x ( t + h ) − x ( t ) | = o ( h α ) uniformly in t ∈ [0 , 1] , Hamadouche [9] established that the sequence ( ξ n pg ) n ≥ 1 converges weakly to B in H oα for every α < 1 / 4 but is not tight in H oα as soon as α ≥ 1 / 4 . In some sense this result means that polygonal smoothing is too violent. With convolution smoothing of ξ n , it is possible to reach the weak Hölder convergence for any α < 1 / 2 , see [10]. This paper investigates the convergence in law of ξ p n g and of χ n pg with re-spect to the more general class of Hölder spaces H oρ of functions x satifying | x ( t + h ) − x ( t ) | = o ( ρ ( h )) uniformly in t ∈ [0 , 1] , where ρ ( h ) = h α L (1 /h ) with 0 < α < 1 and L is slowly varying at inﬁnity and ultimately monotonic. In particular, this covers the case of spaces H oρ built on weight functions ρ ( h ) = h α ` 1 β 1 ( c 1 /h ) . . . ` kβ k ( c k /h ) , where the ` j ’s are j -iterated logarithms. The critical weight function for the Hölder regularity of B being ρ 0 ( h ) = h 1 / 2 ln 1 / 2 ( c/h ) , no stronger topological framework than H oρ 0 can be expected for the conver-gence in law to B of a sequence of polygonal processes. Recent limit theorems in the spaces H oρ may be found in Račkauskas and Suquet [16, 18, 19]. Some statistical applications of weak Hölder convergence are proposed by the same authors [17, 20, 21]. We prove in the present contribution that ξ n pg converges in law to B in H oρ 2

if and only if h 1 / 4 = o ( ρ ( h )) . Hence the weight function ρ ( h ) = h 1 / 4 is really the right critical one in this problem. The polygonal quantile process behaves better with respect to Hölder topologies. Indeed we prove that χ n pg converges in law to B in H oρ if and only if h 1 / 2 ln(1 /h ) = o ( ρ ( h )) . The paper is organized as follows. The relevant background on Hölder spaces and weak convergence therein is presented in Section 2. Section 3 contains the limit theorem for ξ p n g and its proof. Section 4 does the same for χ p n g . 2 Preliminaries 2.1 Polygonal processes Let U 1 , . . . , U n be a sample of i.i.d. random variables uniformly distributed on [0 , 1] . We denote by U n : i the order statistics of the sample 0 = U n :0 ≤ U n :1 ≤ ∙ ∙ ∙ ≤ U n : n ≤ U n : n +1 = 1 , which are distinct with probability one. For notational convenience, put u n : i = E U n : i = n + i 1 , i = 0 , 1 , . . . , n + 1 . We recall the distributional equality (see e.g. [24]) ( U n :1 , . . . , U n : n ) = d  SS n 1+1 , . . . , S n S + n 1  , (1) where S k = X 1 + ∙ ∙ ∙ + X k and the X k ’s are i.i.d 1-exponential random variables. The empirical distribution function F n of the sample is 1 n F n ( t ) := n X 1 { U i ≤ t } . i =1 It is also the piecewise constant function which jumps at the U i ’s and satisﬁes F n ( U n : i ) = ni, 0 ≤ i ≤ n. Note that F n ( U n : n +1 ) = 1 . We deﬁne the polygonal empirical function F n pg as the polygonal random line with vertices  U n : i , F n ( U n : i )  , i = 0 , 1 , . . . , n + 1 . The uniform empirical process ξ n and the polygonal uniform empirical process ξ n pg are deﬁned respectively by ξ n ( t ) := √ n  F n ( t ) − t  , ξ p n g ( t ) := √ n  F n pg ( t ) − t  , t ∈ [0 , 1] . The polygonal smoothing being non linear, F n pg does not inherit of the unbi-asedness of F n . Nevertheless the obvious estimate k ξ n − ξ n pg k ∞ ≤ √ 1 n , (2) 3

implies that ξ n pg , like ξ n , converges in the sense of the ﬁnite dimensional distri-butions to the Brownian bridge B . We deﬁne the (discontinuous) uniform quantile process χ n by n +1 χ n ( t ) = √ n X U n : i 1 ( u n : i − 1 ,u n : i ] ( t ) − t  , t ∈ [0 , 1] . (3) i =1 Deﬁnition (3) diﬀers slightly of the most usual one for the uniform quantile process, see e.g. [5]. This later, denoted here χ ˜ n is given by χ ˜ n ( t ) := √ n  F n − 1 ( t ) − t  , t ∈ [0 , 1] , where F n − 1 ( t ) := inf { u ; F n ( u ) ≥ t } . The advantage of (3) is that at each jump of χ n , the process has null expectation. We associate to χ n the polygonal uniform quantile process χ p n g which is aﬃne on each [ u n : i − 1 , u n : i ] , i = 1 , . . . , n + 1 and satisﬁes χ p n g ( u n : i ) = √ n ( U n : i − u n : i ) , i = 0 , 1 , . . . , n + 1 . (4) We shall also consider the polygonal smoothing χ ˜ p n g of χ ˜ n deﬁned as the polyg-onal line which is aﬃne on each [( i − 1) /n, i/n ] , i = 1 , . . . , n and satisﬁes χ p n g ( i/n ) = √ n ( U n : i − i/n ) , i = 0 , 1 , . . . , n. (5) ˜ 2.2 Hölder spaces Throughout this paper we deal with Hölder spaces H ρ or H oρ built on some weight function ρ satisfying the following condition. (r1) The function ρ : [0 , 1] → R is non decreasing continuous and such that ρ ( h ) = h α L (1 /h ) , 0 < h ≤ 1 , (6) where 0 < α < 1 , and L is some positive function which is normalized slowly varying at inﬁnity. Let us recall that L ( t ) is positive continuous normalized slowly varying at inﬁnity if it has a representation L ( t ) = c exp Z bt ε ( u ) d uu  , with 0 < c < ∞ constant and ε ( u ) → 0 when u → ∞ . By a theorem of Bojanic and Karamata [1, Th.1.5.5], the class of normalized slowly varying functions is exactly the Zygmund class i.e. the class of functions f ( t ) such that for every δ > 0 , t δ f ( t ) is ultimately increasing and t − δ f ( t ) is ultimately decreasing. Here and below “ultimately” means “on some interval [ b, ∞ ) ”. We shall also use one of the following extra assumptions. 4

(r2) The function L in (6) is ultimately monotonic. (r3) The function θ ( t ) := t 1 / 2 ρ (1 /t ) is C 1 on [1 , ∞ ) and there is a β > 1 / 2 , such that θ ( t ) ln − β ( t ) is ultimately non decreasing. Note that such a β always (resp. never) exists when α < 1 / 2 (resp. α > 1 / 2 ). In particular (r3) is satisﬁed by ρ ( h ) = h 1 / 2 ln γ ( c/h ) for γ > 1 / 2 . Here c denotes any constant compatible with the requirement of increasingness of ρ on [0 , 1] . Denote as usual by C[0 , 1] the space of continuous functions x : [0 , 1] → R endowed with the supremum norm k x k ∞ . For ρ satisfying (r1), put − x ( s ) | , 0 < δ ω ρ ( x, δ ) := s,t s ∈ u [0 p , 1] , | x ( ρt () t − s ) ≤ 1 . 0 <t − s<δ We associate to ρ the Hölder space H ρ := { x ∈ C[0 , 1]; ω ρ ( x, 1) < ∞} , endowed with the norm k x k ρ := | x (0) | + ω ρ ( x, 1) . As H ρ is a non separable Banach space, it is more convenient to work with its closed separable subspace H ρo := { x ∈ H ρ ; δ li → m 0 ω ρ ( x, δ ) = 0 } . When ρ ( h ) = h α , the corresponding Hölder spaces H ρ and H ρo will be denoted by H α and H oα respectively. As polygonal lines, the paths of ξ n pg and χ p n g belong clearly to H ρo for any ρ satisfying (r1). One interesting feature of the spaces H oα is the existence of a basis of trian-gular functions, see [3]. We write this basis as a triangular array of functions, indexed by the dyadic numbers. Let us denote by D j the set of dyadic numbers in [0 , 1] of level j , i.e. D 0 = { 0 , 1 } , D j =  (2 l − 1)2 − j ; 1 ≤ l ≤ 2 j − 1  , j ≥ 1 . Write for r ∈ D j , j ≥ 0 , − j r − := r − 2 − j , r + := r + 2 . For r ∈ D j , j ≥ 1 , deﬁne the triangular Faber-Schauder functions Λ r by: Λ r (  2 j ( t − r − ) if t ∈ ( r − , r ] ; t ) := 2 j ( r + − t ) if t ∈ ( r, r + ] ; 0 else.

When j = 0 , we just take the restriction to [0 , 1] in the above formula, so Λ 0 ( t ) = 1 − t, Λ 1 ( t ) = t, t ∈ [0 , 1] . The sequence { Λ r ; r ∈ D j , j ≥ 0 } is a Schauder basis of C[0 , 1] . Each x ∈ C[0 , 1] has a unique expansion ∞ x = X X λ r ( x )Λ r , (7) j =0 r ∈ D j with uniform convergence on [0 , 1] . The Schauder scalar coeﬃcients λ r ( x ) are given by x r ( r − ) λ r ( x ) = x ( r ) − ( + )2+ x , r ∈ D j , j ≥ 1 , and in the special case j = 0 by λ 0 ( x ) = x (0) , λ 1 ( x ) = x (1) . The partial sum J E J x := X X λ r ( x )Λ r (8) j =0 r ∈ D j in the series (7) gives the linear interpolation of x by a polygonal line between the dyadic points of level at most J . Ciesielski [3] proved that { Λ r ; r ∈ D j , j ≥ 0 } is also a Schauder basis of each space H oα (hence the convergence (7) holds in the H α topology when x ∈ H oα ) and that the norm k x k α is equivalent to the following sequence norm : jα max λ r k x k s α eq := s j u ≥ p 0 2 r ∈ D j | ( x ) | . This equivalence of norms provides a very convenient discretization procedure to deal with Hölder spaces and is extended in [18] to the spaces H oρ with ρ satisfying (r1). The sequence norm k x k s ρ eq equivalent to k x k ρ is then deﬁned by k x k s ρ eq := s j u ≥ p 0 ρ (21 j ) r m ∈ a D x j | λ r ( x ) | . (9) It is worth noticing that k x − E J x k s ρ eq = j su > p J ρ (12 j ) r m ∈ a D x j | λ r ( x ) | . (10) 2.3 Tightness in Hölder spaces We write H o Y n −−− ρ −→ Y, n →∞ for the convergence in law in the separable Banach space H oρ of a sequence ( Y n ) n ≥ 1 of random elements in H oρ (also called here convergence in distribution in 6

H oρ or weak convergence in H ρo ). Such a convergence is equivalent to the tightness of ( Y n ) n ≥ 1 on H ρo together with convergence of ﬁnite dimensional distributions. The following characterization of tightness in H ρo looks very similar to the classical one for the space C[0 , 1] , obtained by combining Ascoli’s and Prohorov’s theorems. Theorem 1. The sequence ( Y n ) n ≥ 1 of random elements in H oρ is tight if and only if the following two conditions are satisﬁed: i) For each t ∈ [0 , 1] , the sequence ( Y n ( t )) n ≥ 1 is tight on R . ii) For each ε > 0 , δ li → m 0 lim sup P ( ω ρ ( Y n , δ ) > ε ) = 0 . (11) n →∞ Proof sketched. For the suﬃciency, we refer to Theorem 3 in [18] noting that with the notations used therein, k Y n − E j Y n k ρ seq ≤ ω ρ ( Y n , 2 − j ) . For the necessity, introduce the functionals Φ N deﬁned on H ρo by Φ N ( x ) := ω ρ  x, 1 /N  . By the deﬁnition of H oρ , the sequence (Φ N ) N ≥ 1 decreases to zero pointwise on H oρ . Moreover each Φ N is continuous in the strong topology of H ρo . By Dini’s theorem this gives the uniform convergence of (Φ N ) N ≥ 1 to zero on any compact K of H ρo . This remark combined with the assumption of tightness of ( Y n ) n ≥ 1 leads easily to lim s N →∞ n u ≥ p 1 P ( ω ρ ( Y n , 1 /N ) > ε ) = 0 , (12) from which we obtain (11). The special following tightness result extends Theorem 1 in [9]. It may be relevant in the case of processes Y n whose “not too small increments” behave smoothly, while their smaller increments can be controlled diﬀerently. This typically happens with the empirical processes ξ p n g . Theorem 2. Assume that the sequence ( Y n ) n ≥ 1 of random elements in H ρo fulﬁls the following conditions. a) For each r ∈ D ,  Y n ( r )  n ≥ 1 is tight in R . b) For some non increasing sequence ( a n ) n ≥ 1 in (0 , 1) , converging to zero, P  | Y n ( t ) − Y n ( s ) | ≥ u  ≤ | t − s | Q ( | t − s | , u ) , u > 0 , | t − s | ≥ a n , (13) where the function Q : R ∗ + × R ∗ + → R + satisﬁes for every positive ε , + ∞ X Q  2 − j , ερ (2 − j )  < ∞ . (14) j =1 7

c) ω ρ ( Y n , a n ) converges to 0 in probability ( n → ∞ ). Then ( Y n ) n ≥ 1 is tight in H oρ . When using Theorem 2 to prove a weak Hölder convergence, Condition a) is automatically satisﬁed as soon as the convergence of ﬁnite dimensional distri-butions of Y n holds true. Note also that if b) is satisﬁed without the restriction | t − s | ≥ a n , then c) follows. Of course our interest in this theorem focuses on the case where there is no possibility to obtain b) without the restriction | t − s | ≥ a n . Condition c) may be tractable when some information is available on the local smoothness of Y n . The following corollary is well adapted to the case of the polygonal uniform empirical process. Recall that ρ ( h ) = h α L (1 /h ) with 0 < α < 1 and L slowly varying. Corollary 3. Assume that the sequence ( Y n ) n ≥ 1 of random elements in H oρ satisﬁes Conditions a) and c) of Theorem 2 and that for some real numbers γ > 0 , p > 2 and some non increasing sequence ( a n ) n ≥ 1 in (0 , 1) , converging to zero, E | Y n ( t ) − Y n ( s ) | p ≤ C p | t − s | 1+ γ , | t − s | ≥ a n , (15) for some positive constant C p . Then ( Y n ) n ≥ 1 is tight in H oρ if either α < γ/p or α = γ/p with P j ≥ 1 L (2 j ) − p < ∞ . Proof of Theorem 2. By Theorem 2 and Remark 1 in [19], it suﬃces to prove that for every positive ε , lim ∞ li n m sup P  k Y n − E J Y n k ρ seq > ε  = 0 , (16) J →→∞ with the projectors E J deﬁned by (8). Deﬁne the integer J n by the condition 2 − J n − 1 < a n ≤ 2 − J n . Then accounting (10), we have for each J ≥ 1 P  k Y n − E J Y n k ρ seq > ε  ≤ P 0 J,n + P n 00 , (17) where 0 P J,n := P  J ≤ m j a ≤ x J n ρ (21 − j ) r m ∈ a D x j | λ r ( Y n ) | > ε  , 1 P n 00 := P  j s > u J p n ρ (2 − j ) r m ∈ a D x j | λ r ( Y n ) | > ε  . The estimate (17) is clear when J ≤ J n . When J n < J , it remains true with the usual convention “ sup ∅ = −∞ ”, which gives P 0 J,n = 0 . To control P n 00 , let us simply note that P n 00 ≤ P  ω ρ ( Y n , a n ) > ε  , so Condition c) in Theorem 2 gives lim sup P n 00 = 0 . n →∞ 8

(18)

Now to control P 0 J,n , having in mind the usual convention that a sum indexed by the emptyset is deﬁned as null, we get by Condition b) P 0 J,n ≤ X X P  | λ r ( Y n ) | > ερ (2 − j )  J ≤ j ≤ J n r ∈ D j ≤ X 2 j 2 − j Q  2 − j , ερ (2 − j )  J ≤ j ≤ J n ∞ ≤ X Q  2 − j , ερ (2 − j )  . j = J In view of (14), this leads to J lim lim sup P n 00 = 0 . (19) →∞ n →∞ Then (16) and hence the tigthness of ( Y n ) follow from (18) and (19). Proof of Corollary 3. By Markov’s inequality, (15) leads to (13) with Q ( v, u ) := C p u γ v − p . Hence Condition (14) reduces to ∞ X 2 j ( pα − γ ) j =1 L (2 j ) p < ∞ . Since L is slowly varying in the neighbourhood of inﬁnity, z δ L ( z ) p goes to inﬁnity with z for any positive δ . It follows that the above series converges for any α < γ/p whatever the choice of the slowly varying function L may be. 2.4 An Hölderian FCLT We shall need the following invariance principle for partial sums processes, which is proved in [18] in a more general setting. Let X 1 , . . . , X n , . . . be i.i.d. random variables in R with null expectation and E X 12 = 1 . Set S 0 = 0 , S k = X 1 + ∙ ∙ ∙ + X k , for k = 1 , 2 , . . . and consider the polygonal partial sums processes Ξ n ( t ) = n − 1 / 2 S [ nt ] + n − 1 / 2 ( nt − [ nt ]) X [ nt ]+1 , t ∈ [0 , 1] . (20) Theorem 4 (Račkauskas, Suquet [18]). Let ρ satisfying (r1). If Ξ n con-verges weakly in H ρo to the standard Brownian motion W , then for every A > 0 , t l → im ∞ t P  | X 1 | ≥ Aθ ( t )  = 0 . (21) If ρ satisﬁes (r1), (r3) and (21), then Ξ n converges weakly in H ρo to W .

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