1Central limit the o rems in Holder spaces
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1Central limit the- o- rems in Holder spaces Random fields and central limit theorem in some generalized Holder spaces A. RACˇKAUSKAS?, CH. SUQUET? Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania Laboratoire de Statistique et Probabilites, E.P. CNRS 1765 Bat. M2, Cite Scientifique, F-59655 Villeneuve d'Ascq Cedex, France Abstract. For rather general moduli of smoothness ? (like e.g. ?(h) = h? ln?(c/h) ) the Holder spaces H?([0, 1]d), are characterized by the rate of coefficients in the skew pyramidal basis. With this analytical tool, we study in terms of second differences the existence of a version in H? for a given random field. In the same spirit, central limit theorems are obtained both for i.i.d. and martingale differences sequences of random elements in H?. 1. INTRODUCTION In many situations, stochastic processes and random fields have a smoothness intermediate between the continuity and differentiability. The scale of Holder spaces is then a natural functional framework to investigate the regularity of such processes and fields. And weak convergence in this setting is a stronger result than in the space of continuous functions. In this paper we consider the scale of generalized Holder spaces H?([0, 1]d), where ? is a modulus of smoothness (precise definition is given in Section 1 below) and discuss two questions: (I) For a given random field indexed by [
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1Central limit the- o- rems in Holder spaces Random fields and central limit theorem in some generalized Holder spaces A. RACˇKAUSKAS?, CH. SUQUET? Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania Laboratoire de Statistique et Probabilites, E.P. CNRS 1765 Bat. M2, Cite Scientifique, F-59655 Villeneuve d'Ascq Cedex, France Abstract. For rather general moduli of smoothness ? (like e.g. ?(h) = h? ln?(c/h) ) the Holder spaces H?([0, 1]d), are characterized by the rate of coefficients in the skew pyramidal basis. With this analytical tool, we study in terms of second differences the existence of a version in H? for a given random field. In the same spirit, central limit theorems are obtained both for i.i.d. and martingale differences sequences of random elements in H?. 1. INTRODUCTION In many situations, stochastic processes and random fields have a smoothness intermediate between the continuity and differentiability. The scale of Holder spaces is then a natural functional framework to investigate the regularity of such processes and fields. And weak convergence in this setting is a stronger result than in the space of continuous functions. In this paper we consider the scale of generalized Holder spaces H?([0, 1]d), where ? is a modulus of smoothness (precise definition is given in Section 1 below) and discuss two questions: (I) For a given random field indexed by [
- gaussian process
- ho? always
- banach isomorphism between
- analytical tool
- smoothness ?
- central limit theorems
- differences sequences
- odic function
- functions
Central limit the-o-rems in H¨older spaces
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2 The usefulness of Δ h 2 ξ ( t ) in the problem of sample paths differentiability is known(seeCram´er-Leadbetter(1967)).Fromananalyticalpointofview, there is no loss in working with Δ h 2 f tostudytheHo¨lderregularityofa non-random function f. This observation goes back to Zygmund (1945) who noticed that a necessary and sufficient condition that a continuous and peri-odic function f ( x )shouldsatisfyaHo¨lderconditionoforder α, 0 < α < 1 , is that Δ 2 h f ( x ) = O ( h α ) , as h → +0 , uniformly in x. The role of Δ h 2 is now well understood in the more general context of Besov spaces (see Peetre (1976)). From a probabilistic point of view, it is clear that any control in probability on Δ h 1 ξ ( t ) provides similar type of control on Δ 2 h ξ ( t ) , but the converse is false in general. So the use of Δ 2 h ξ ( t ) , brings more flexibility in our basic assumptions. Moreover the second difference appear very naturally in the discretization procedure corresponding to the decomposition of a function in the Faber Schauder basis of triangular functions (those obtained by means of the affine interpolation between dyadic points). Thepresentcontributionextendsourpreviousresults(RacˇkauskasansSuquet (1998)) in the following three directions • use of more general moduli of smoothness ρ ; • multidimensional parameter space [0 , 1] d ; • central limit theorem for martingale differences. To this end we follow the procedure already used in the dimension 1 , replacing the basis of triangular functions by some special basis for pyramidal functions. It turns out that the coefficients in this basis are the second differences and we have a very convenient Banach isomorphism between H ρ and appropriate sequences space. This analytical background is detailed in Section 2. In Section 3 we obtain sufficient conditions for the existence of a H ρ version of a given random field. In Section 4 we give central limit theorems in H ρ in the i.i.d. case and also for triangular arrays of martingale differences. 2. ANALYTICAL BACKGROUND Throughout T = [0 , 1] d and R d is endowed with the norm | t | := 1 m ≤ i a ≤ x d | t i | , t = ( t 1 , . . . , t d ) ∈ R d . Denote by H ρ the set of real valued continuous functions x : T → R such that w ρ ( x, 1) < ∞ , where w ρ ( x, δ ) := sup | x ( t ) − x ( s ) | t,s ∈ T , 0 < | t − s | <δ ρ ( | s − t | )
3 and ρ is a modulus of smoothness satisfying conditions (1) to (5) below where c 1 , c 2 and c 3 are positive constants:
ρ (0) = 0 , ρ ( δ ) > 0 , 0 < δ ≤ 1; (1) ρ is non decreasing on [0 , 1]; (2) ρ (2 δ ) ≤ c 1 ρ ( δ ) , 0 ≤ δ ≤ 1 / 2; (3) Z 0 δ ρ ( uu ) du ≤ c 2 ρ ( δ ) , 0 < δ ≤ 1; (4) δ Z δ 1 ρ ( u 2 u ) du ≤ c 3 ρ ( δ ) , 0 < δ ≤ 1 . (5) For instance, elementary computations show that the functions ρ ( δ ) := δ α ln β δc , 0 < α < 1 , β ∈ R , satisfy conditions (1) to (5), for a suitable choice of the constant c, namely c ≥ exp( β/α ) if β > 0 and c > exp( − β/ (1 − α )) if β < 0. The set H ρ is a Banach space when endowed with the norm k x k ρ := | x (0) | + w ρ ( x, 1) . Obviously an equivalent norm is obtained replacing | x (0) | in the above formula by k x k ∞ := sup {| x ( t ) | ; t ∈ T } . Define H ρo = { x ∈ H ρ ; δ li → m 0 w ρ ( x, δ ) = 0 } . Then H ρo is a closed subspace of H ρ . Now let us remark that for any function ρ satisfying (1)–(5), there is a positive constant c 4 such that ρ ( δ ) ≥ c 4 δ, 0 ≤ δ ≤ 1 . (6) Hence the spaces H ρo always contain all the Lipschitz functions and in par-ticular the (continuous) piecewise affine functions. The separability of the spaces H ρo follows by standard interpolation arguments. Since we are interested in the analysis of these spaces in terms of second differences of the functions x, our first task is to establish the equivalence of the norm k x k ρ with some sequential norm involving the dyadic second differences of x . To this aim, we shall use some Schauder basis of pyramidal functions. Our main reference for this part is Semadeni (1982). The so called skew pyramidal basis was introduced by Bonic Frampton and Tromba (1969) and independently by Ciesielski and Geba (see the historical notes in
4 Semadeni (1982) p. 72). This choice, which is not the only possible, leads to rather simple formulas for the Schauder coefficients in terms of second differences. To explain the construction of the skew pyramidal basis, consider first a cube Q = s + aT = { s + X u i e i ; 0 ≤ u i ≤ a } , 1 ≤ i ≤ d where the e i ’s denote the vectors of the canonical basis of R d . The standard triangulation of Q is the family T ( Q ) of simplexes defined as follows. Write Π d for the set of permutations of the indexes 1 , . . . , d . For any π = ( i 1 , . . . , i d ) ∈ Π d , let Δ π ( Q ) be the convex hull of the d + 1 points d s, s + ae i 1 , s + a ( e i 1 + e i 2 ) , . . . , s + a X e i k . k =1 So, each simplex Δ π ( Q ) corresponds to one path from s to s 0 = s + a (1 , . . . , 1) via vertices of Q and such that along each segment of the path, only one coor-dinate increases while the others remain constants. Thus Q is divided into d ! simplexes with disjoint interiors. Next let H i be the hyperplane perpendicular to e i and passing through the middle of the edge [ s, s + ae i ]. The hyperplanes H 1 , . . . , H d divide the cube Q into 2 d cubes, say, Γ k ( Q ) , k = 0 , . . . , 2 d − 1. More precisely, if k = ε 1 + ε 2 2 1 + . . . + ε d 2 d − 1 is the binary representation of k, Γ k ( Q ) := s +21 X d ε i e i + aT. 2 i =1 By lemma 3.4.2 in Semadeni (1982), each simplex Δ π (Γ k ( Q )) of the stan-dard triangulation of Γ k ( Q ) is contained in an unique simplex Δ π 0 ( Q ) of the standard triangulation of Q . Consider now the sequence ( P j ) j ≥ 0 of partitions of T defined by P 0 := { T } , P j := { Γ k ( Q ); Q ∈ P j − 1 , 0 ≤ k < 2 d } . In other words, P j is composed of the 2 jd cubes obtained by dividing each edge of the cube [0 , 1] d into 2 j segments of length 2 − j . Finally we define the triangulation T j as the union of the standard triangulations of the cubes in P j . T j := { Δ π ( Q ); Q ∈ P j , π ∈ Π d } , j = 0 , 1 , . . . Clearly the set W j := vert( T j ) of vertices of the simplexes in T j is the set of vertices of the cubes in P j , whence W j = vert( T j ) = { k 2 − j ; 0 ≤ k ≤ 2 j } d .
5 In what follows we put V 0 := W 0 and V j := W j \ W j − 1 for j ≥ 1. So V j is the set of new vertices born with the triangulation T j . More explicitly, V j is the set of dyadic points v = ( k 1 2 − j , . . . , k d 2 − j ) in W j with at least one k i odd. The T j -pyramidal function Λ j,v with peak vertex v ∈ V j is defined on T by the three conditions i) Λ j,v ( v ) = 1; ii) Λ j,v ( w ) = 0 if w ∈ vert( T j ) and w 6 = v ; iii) Λ j,v is affine on each simplex Δ in T j , i.e., if the w i are the vertices of Δ , d d d Λ j,v X r i w i = X r i Λ j,v ( w i ) , r i ≥ 0 , X r i = 1 . i =0 i =0 i =0 From iii) it follows clearly that the support of Λ j,v is the union of all simplexes in T j containing the peak vertex v. By Proposition 3.4.5 in Semadeni (1982), the functions Λ j,v are obtained by dyadic translations and changes of scale: Λ j,v ( t ) = Λ(2 j ( t − v )) , t ∈ T , v ∈ V j from the same function Λ with support included in [ − 1 , 1] d : Λ( t ) := max 0 , 1 − t m i a < x 0 | t i | − t m i > a 0 x t i , t = ( t 1 , . . . , t d ) ∈ [ − 1 , 1] d . But this apparent simplicity is misleading. The edges effects due to the re-striction to t ∈ T give different shapes for the supports of the Λ j,v ’s. For instance when d = 2 , the support of Λ is hexagonal, but among the five func-tions Λ 1 ,v , only one has hexagonal support (corresponding to the peak vertex v = (1 / 2 , 1 / 2)), the four others having pentagonal supports. The skew pyramidal basis is the family L := { Λ j,v ; j ≥ 0 , v ∈ V j } lexico-graphically ordered. As a special case of the Proposition 3.1.6. in Semadeni (1982), L is a Schauder basis of the Banach space C ( T ) of real valued con-tinuous functions on T . Hence any x ∈ C ( T ) admits the unique uniformly convergent series expansion: ∞ x ( t ) = X X λ j,v ( x )Λ j,v ( t ) , t ∈ T . j =0 v ∈ V j The Schauder coefficients λ j,v ( x ) are given by: λ 0 ,v ( x ) = x ( v ) , v ∈ V 0 ; λ j,v ( x ) = x ( v ) − 12 x ( v − ) + x ( v + ) , v ∈ V j , j ≥ 1 .
