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Necessary and sufficient condition for the functional centrallimittheoreminH¨olderspaces ByAlfredasRacˇkauskas 1 , 3 , 4 and Charles Suquet 2 , 3 Revised version September 10, 2003
Abstract Let ( X i ) i 1 be an i.i.d. sequence of random elements in the Banach space B , S n := X 1 + ∙ ∙ ∙ + X n and ξ n be the random polygonal line with vertices ( k/n, S k ), k = 0 , 1 , . . . , n . Put ρ ( h ) = h α L (1 /h ), 0 h 1 with 0 < α 1 / 2 and L slowly varying at infinity. Let H oρ ( B ) be the H¨olderspaceoffunctions x : [0 , 1] 7→ B , such that || x ( t + h ) x ( t ) || = o ( ρ ( h )), uniformly in t . We characterize the weak convergence in H ρo ( B ) of n 1 / 2 ξ n to a Brownian motion. In the special case where B = R and ρ ( h ) = h α , our necessary and sufficient conditions for such convergence are E X 1 = 0 and P ( | X 1 | > t ) = o ( t p ( α ) ) where p ( α ) = 1 / (1 / 2 α ). This completes Lamperti (1962) invariance principle. MSC 2000 subject classifications . Primary-60F17; secondary-60B12. Key words and phrases .CentrallimittheoreminBanachspaces,Ho¨lder space, invariance principle, partial sums process.
1 Department of Mathematics, Vilnius University, Naugarduko 24, Lt-2006 Vilnius, Lithuania. E-mail: Alfredas.Rackauskas@maf.vu.lt 2 CNRSFRE2222,LaboratoiredeMath´ematiquesApplique´es,Bˆat.M2, Universit´eLilleI,F-59655VilleneuvedAscqCedex,France. E-mail: Charles.Suquet@univ-lille1.fr 3 Research supported by a cooperation agreement CNRS/LITHUANIA (4714). 4 Partially supported by Vilnius Institute of Mathematics and Informatics. 1
A.RacˇkauskasandCh.Suquet
1 Introduction Let ( B, k k ) be a separable Banach space and X 1 , . . . , X n , . . . be i.i.d. random elements in B . Set S 0 = 0, S k = X 1 + ∙ ∙ ∙ + X k , for k = 1 , 2 , . . . and consider the partial sums processes ξ n ( t ) = S [ nt ] + ( nt [ nt ]) X [ nt ]+1 , t [0 , 1] and ξ s n r := n 1 / 2 ξ n . In the familiar case where B is the real line R , classical Donsker-Prohorov invariance principle states, that if E X 1 = 0 and E X 12 = 1, then sr D ξ n W, (1) in C [0 , 1], where ( W ( t ) , t R ) is a standard Wiener process and D denotes convergence in distribution. The finiteness of the second moment of X 1 is clearly necessary here, since (1) yields that ξ n sr (1) satisfies the central limit theorem. Replacing C [0 , 1] in (1) by a stronger topological framework provides more continuous functionals of paths. With this initial motivation, Lamperti [7] considered the convergence (1)withrespecttosomeHo¨lderiantopology.Letusrecallhisresult. For 0 < α < 1, let H oα be the vector space of continuous functions x : [0 , 1] R such that l δ i m 0 ω α ( x, δ ) = 0, where ω α ( x, δ ) = sup | x ( | tt ) sx | ( α s ) | . s,t [0 , 1] , 0 <t s<δ H oα is a separable Banach space when endowed with the norm k x k α := | x (0) | + ω α ( x, 1) . Lamperti [7] proved that if 0 < α < 1 / 2 and E | X 1 | p < , where p > p ( α ) := 1 / (1 / 2 α ), then (1) takes place in H oα . This result was derived again by Kerkyacharian and Roynette [5]byanothermethodbasedonCiesielski[2]analysisofHo¨lderspacesbytriangular functions. Further generalizations were given by Erickson [3] (partial sums processes indexed by [0 , 1] d ), Hamadouche [4] (weakly dependent sequence ( X n )),Racˇkauskasand Suquet [10] (Banach space valued X i sandH¨olderspacesbuiltonthemoduli ρ ( h ) = h α ln β (1 /h )). Considering a symmetric random variable X 1 such that P { X 1 u } = cu p ( α ) , u 1, Lamperti [7] noticed that the sequence ( ξ s n r ) is not tight in H oα . This gives some hint that thecostoftheextensionoftheinvarianceprincipletotheHo¨lderiansettingisbeyondthe square integrability of X 1 . The simplest case of our general result provides a full answer to this question for the space H oα . Theorem 1. Let 0 < α < 1 / 2 and p ( α ) = 1 / (1 / 2 α ) . Then ξ n sr D −→ W in the space H oα n →∞ if and only if E X 1 = 0 and lim t p ( α ) P {| X 1 | ≥ t } = 0 . t →∞ 2
(2)
FCLTinH¨olderspaces
WewouldliketopointherethatTheorem1contrastsstronglywiththeH¨olderian invariance principle for the adaptive self-normalized partial sums processes ζ s n e . These are defined as random polygonal lines of interpolation between the vertices ( V k 2 /V n 2 , S k /V n ), k = 0 , 1 , . . . , n , where V 02 = 0 and V k 2 = X 12 + . . . X k 2 . It is shown in [11] that ( ζ s n e ) converges in distribution to W in any H oα (0 < α < 1 / 2) provided that E | X 1 | 2+ ε is finite for some arbitrary small ε > 0. This condition can even be relaxed into “ X 1 is in the domain of attraction of the normal distribution” in the case of symmetric X i ’s (this last condition is also necessary). To describe our general result, some notations are needed here. We write C( B ) for the Banach space of continuous functions x : [0 , 1] B endowed with the supremum norm k x k := sup {k x ( t ) k ; t [0 , 1] } . Let ρ be a real valued non decreasing function on [0 , 1], null and right continuous at 0, positive on (0 , 1]. Put ω ρ ( x, δ ) := sup k x ( ρt () t xs () s ) k . s,t [0 , 1] , 0 <t s<δ We associate to ρ theHo¨lderspace H oρ ( B ) := { x C( B ); l δ im 0 ω ρ ( x, δ ) = 0 } , equiped with the norm k x k ρ := k x (0) k + ω ρ ( x, 1) . We say that X 1 satisfies the central limit theorem in B , which we denote by X 1 CLT( B ), if n 1 / 2 S n converges in distribution in B . This implies that E X 1 = 0 and X 1 is pregaussian . It is well known (e.g. [8]), that the central limit theorem for X 1 cannot be characterized in general in terms of integrability of X 1 and involves the geometry of the Banach space B . Of course some integrability of X 1 and the partial sums is necessary for the CLT. More precisely, e.g. [8, Corollary 10.2], if X 1 CLT( B ), then lim t 2 sup P k S n k > t n = 0 . t →∞ n 1 The space CLT( B ) may be endowed with the norm clt ( X 1 ) := sup E k n 1 / 2 S n k . (3) n 1 Let us recall that a B valued Brownian motion W with parameter µ ( µ being the distribu-tion of a Gaussian random element Y on B ) is a Gaussian process indexed by [0 , 1], with independent increments such that W ( t ) W ( s ) has the same distribution as | t s | 1 / 2 Y . The extension of the classical Donsker-Prohorov invariance principle to the case of B -valued partial sums is due to Kuelbs [6] who established that ξ n sr converges in distribution in C( B ) to some Brownian motion W if and only if X 1 CLT( B ). This convergence of ξ s n r will be referred to as the functional central limit theorem in C( B ) and denoted by X 1 FCLT( B ). Of course in Kuelbs FCLT, the parameter µ of W is the Gaussian distribution on B with same expectation and covariance structure as X 1 . The stronger property of convergence in distribution of ξ s n r in H oρ ( B ) will be denoted by X 1 FCLT( B, ρ ). An obvious preliminary requirement for the FCLT in H ρo ( B ) is that the B -valued Brownian motion has a version in H oρ ( B ). From this point of view, the critical ρ is ρ c ( h ) = p h ln(1 /h )duetoLe´vysTheoremonthemodulusofuniformcontinuityofthe Brownian motion (see e.g. [12] and Proposition 4 below). So our interest will be restricted 3
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to functions ρ generatingaweakerHo¨ldertopologythan ρ c . More precisely, we consider the functions ρ of the form ρ ( h ) = h α L (1 /h ) where 0 < α 1 / 2 and L is slowly varying at infinity. Moreover when α = 1 / 2, we assume that L ( t ) increases faster at infinity than ln β t for some β > 1 / 2. Throughout the paper we use the notation θ ( t ) = t 1 / 2 ρ t 1 , t 1 . (4) OurcharacterizationoftheFCLTintheHo¨lderspaceH oρ ( B ) reads now simply: X 1 FCLT( B, ρ ) if and only if X 1 CLT( B ) and for every A > 0, t li m t P k X 1 k ≥ ( t ) = 0 . Moreover when α < 1 / 2, it is enough to take A = 1 in the above condition. Clearly in the special case B = R and ρ ( h ) = h α , this characterization is exactly Theorem 1. It is also worth noticing that like in Kuelbs FCLT, all the influence of the geometry of the Banach space B is absorbed by the condition X 1 CLT( B ). The paper is organized as follows. Section 2 presents some background on the sequen-tialnormequivalenttotheinitialH¨oldernormofH ρo ( B ), the tightness in H oρ ( B ) and the admissibleHo¨ldertopologiesfortheFCLT.Section3givesageneralnecessarycondition which holds even for more general ρ . Section 4 contains the proof of the sufficient part in thecharacterizationofHo¨lderianFCLT.Sometechnicalauxilliaryresultsaredeferredin Section 5 to avoid overweighting of the exposition. 2 Preliminaries 2.1 Analytical background With the aim to use a sequential norm equivalent to k x k ρ , we require, following Ciesielski (see e.g. [13, p.67]), that the modulus of smoothness ρ satisfies the conditions: ρ (0) = 0 , ρ ( h ) > 0 , 0 < h 1; (5) ρ is non decreasing on [0 , 1]; (6) ρ (2 h ) c 1 ρ ( h ) , 0 h 1 / 2; (7) Z h ρ ( uu )d u c 2 ρ ( h ) , 0 < h 1; (8) 0 h Z h 1 ρ ( uu 2 )d u c 3 ρ ( h ) , 0 < h 1; (9) where c 1 , c 2 and c 3 are positive constants. Let us observe in passing, that (5), (6) and (8) together imply the right continuity of ρ at 0. The class of functions ρ satisfying these requirements is rich enough according to the following. Proposition 2. For any 0 < α < 1 , consider the function ρ ( h ) = h α L (1 /h ) where L is normalized slowly varying at infinity, continuous and positive on [1 , ) . Then ρ fulfills conditions (5) to (9) up to a change of scale. 4
¨ FCLT in Holder spaces
Proof. Let us recall that L ( t ) is a 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. It follows that for some 0 < a 1, ρ is non decreasing on [0 , a ]. Then (6) is satisfied by ρ ˜( h ) := ρ ( ah ). Due to the continuity and positivity of ρ ˜ on (0 , 1], each inequality (7) to (9) will be fulfilled if its left hand side divided by ρ ˜( h ) has a positive limit when h goes to 0. For (7), α this limit is clearly 2 . For (8), we have by [1, Prop. 1.5.10], ˜ 1 Z 0 h ρ ( uu )d u = a α Z v α L ( v/a ) d v α 1 ρ ˜( h ) . 1 /h Similarly for (9), we obtain by [1, Prop. 1.5.8], h Z h 1 ρ ˜( uu 2 )d u = a α h Z 11 /h v α L ( v/a ) d v 1 ρ ˜( h ) α .
Write D j for the set of dyadic numbers of level j in [0 , 1], i.e. D 0 = { 0 , 1 } and for j 1, D j = { (2 k + 1)2 j ; 0 k < 2 j 1 } . For any continuous function x : [0 , 1] B , define λ 0 ,t ( x ) := x ( t ) , t D 0 and for j 1, λ j,t ( x ) := x ( t ) 21 x ( t + 2 j ) + x ( t 2 j ) , t D j . The λ j,t ( x ) are the B -valued coefficients of the expansion of x in a series of triangular functions. The j -th partial sum E j x of this series is exactly the polygonal line interpolating x between the dyadic points k 2 j (0 k 2 j ). Under (5) to (9), the norm k x k ρ is equivalent (see e.g. [12]) to the sequence norm k x k ρ seq := s j u 0 p ρ (21 j ) t m a D x j k λ j,t ( x ) k . It is easy to check that k x E j x k s ρ eq = s i> u j p ρ (21 i ) t m a D x i k λ i,t ( x ) k . 5
(10)
A.RaˇckauskasandCh.Suquet
2.2 Tightness The dyadic affine interpolation which is behind the sequential norm is also useful to investigate the tightness in H ρo ( B ). Indeed it is not difficult to check that H oρ ( B ) can be expressed as a topological direct sum of closed subspaces (a Schauder decomposition) by H o ( B ) = M W i . ρ i =0 Here W 0 is the space of B -valued functions defined and affine on [0 , 1] and for i 1, W i is the space of B -valued polygonal lines with vertices at the dyadics of level at most i and vanishing at each dyadic of level less than i . It may be helpful to note here that each W i has infinite dimension with B . This Schauder decomposition of H oρ ( B ) allows us to apply Theorem 3 in [14] and obtain the following tightness criterion. Theorem 3. The sequence ( Y n ) of random elements in H oρ ( B ) is tight if and only if the following two conditions are satisfied: i) For each dyadic t [0 , 1] , the sequence ( Y n ( t )) n 1 is tight on B . ii) For each ε > 0 , j li m lim sup P {k Y n E j Y n k ρ seq > ε } = 0 . (11) n →∞ 2.3AdmissibleH¨oldernorms Let us discuss now the choice of the functions ρ for wich it is reasonable to investigate aHo¨lderianFCLT.If X 1 FCLT( B, ρ ) and ` is a linear continuous functional on B then clearly ` ( X 1 ) FCLT( R , ρ ). So we may as well assume B = R in looking for a r necessary condition on ρ . As polygonal lines, the paths of ξ n s belong to H oρ for any ρ such that h/ρ ( h ) 0, when h 0. The weaker smoothness of the limit process W and the necessity of its membership in H ρo put a more restrictive condition on ρ . Proposition 4. Assume that for some X 1 , the corresponding process ξ s n r converges weakly to W in H ρo . Then lim θ ( t ) t →∞ ln 1 / 2 t = . (12) Proof. Let ω ( W, δ ) denote the modulus of uniform continuity of W . Since W has nec-essarily a version in H oρ , we see that ω ( W, δ ) ( δ ) goes a.s. to zero when δ 0. This convergence may be recast as ( W, δ ) p δ ln 1 = δ li m 0 p ωδ ln(1 ) ρ ( δ () ) 0 a.s. ByL´evysresult[9,Th.52,2]onthemodulusofuniformcontinuityof W , we have with positive probability lim inf δ 0 ω ( W, δ ) / p δ ln(1 ) > 0, so the above convergence implies δ li m 0 p δ ln δ ()1 )0 , = ρ (
which is the same as (12).
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