Necessary and sufficient condition for the functional central limit theorem in Holder spaces

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Necessary and sufficient condition for the functional central limit theorem in Holder spaces By Alfredas Racˇkauskas1,3,4 and Charles Suquet2,3 Revised version September 10, 2003 Abstract Let (Xi)i≥1 be an i.i.d. sequence of random elements in the Banach space B, Sn := X1+· · ·+Xn and ?n be the random polygonal line with vertices (k/n, Sk), k = 0, 1, . . . , n. Put ?(h) = h?L(1/h), 0 ≤ h ≤ 1 with 0 < ? ≤ 1/2 and L slowly varying at infinity. Let Ho?(B) be the Holder space of functions x : [0, 1] 7? B, such that ||x(t+ h)? x(t)|| = o(?(h)), uniformly in t. We characterize the weak convergence in Ho?(B) of n?1/2?n to a Brownian motion. In the special case where B = R and ?(h) = h?, our necessary and sufficient conditions for such convergence are EX1 = 0 and P(|X1| > t) = o(t?p(?)) where p(?) = 1/(1/2 ? ?).

  • general ?

  • self-normalized partial

  • holder spaces

  • sums processes

  • banach space

  • valued coefficients

  • invariance principle

  • x1

  • then ? fulfills


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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)
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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).
6
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3AgeneralrequirementfortheHo¨lderianFCLT We prove now that a necessary condition for X 1 tosatisfytheH¨olderianFCLTinH oρ ( B ) is that for every A > 0, t li m t P X 1 > Aθ ( t ) = 0 . In fact, the same tail condition must hold uniformly for the normalized partial sums, so the above convergence is a simple by-product of the following general result. We point out that Conditions (6) to (9) are not involved here. In this section the restriction on ρ comes from Proposition 4. Theorem 5. If the sequence ( ξ s n r ) n 1 is tight in H oρ ( B ) , then for every positive constant A , t li m t m su p 1 P S m > m 1 / 2 ( t ) = 0 . (13) Proof. As a preliminary step, we claim and check that N li m n sup P ω ρ ξ n sr , 1 /N A = 0 . (14) 1 From the tightness assumption, for every positive ε there is a compact subset K in H oρ ( B ) such that P ω ρ ξ s n r , 1 /N A P ω ρ ξ s n r , 1 /N A and ξ s n r K + ε. Define the functionals Φ N on H ρo ( B ) by Φ N ( f ) := ω ρ f ; 1 /N . By the definition of H ρo ( B ), the sequence (Φ N ) N 1 decreases to zero pointwise on H ρo ( B ). Moreover each Φ N is continuous in the strong topology of H oρ ( B ). By Dini’s theorem this gives the uniform convergence of (Φ N ) N 1 to zero on the compact K . Then we have sup f K Φ N ( f ) < A for every N bigger than some N 0 = N 0 ( A, K ). It follows that for N > N 0 and n 1, P ω ρ ξ s n r , 1 /N A and ξ n sr K = 0 , which leads to P ω ρ ξ sr 1 /N A < ε, N > N 0 , n 1 , n , completing the verification of (14). In particular we get lim sup P ω ρ ξ s m r N , 1 / N →∞ m 1 N A = 0 . (15) Now we observe that sr 1 m k a x N ρ (11 /N ) ξ s m r N ( k/N ) ξ mN (( k 1) /N ) ω ρ ξ s m r N , 1 /N . Writing for simplicity Y k,m = m 1 / 2 ( S mk S m ( k 1) ) , we have from (15) that N li m m su p 1 P n 1 m k a x N Y k,m > Aθ ( N ) o = 0 , (16) recalling that θ ( N ) = N 1 / 2 ρ (1 /N ). By independence and identical distribution of the X ’s, i P n 1 m k a x N Y k,m > Aθ ( N ) o = 1 1 P Y 1 ,m > Aθ ( N )  N . (17) 7
A.RacˇkauskasandCh.Suquet
Consider the function g N ( u ) := 1 (1 u ) N , 0 u 1. As g N is increasing on [0 , 1], we have g N ( u ) g N (1 /N ) = 1 (1 1 /N ) N > 1 e 1 , 1 /N u 1 . (18) By concavity of g N , we also have g N ( u ) N g N (1 /N ) u N (1 e 1 ) u, 0 u 1 /N. (19) Write u m,N := P Y 1 ,m > Aθ ( N ) and u N := sup m 1 u m,N . By increasingness and continuity of g N , sup m 1 g N ( u m,N ) = g N ( u N ). This together with (16) and (17) shows that lim N →∞ g N ( u N ) = 0. By (18), it follows that 0 u N 1 /N , for N large enough. In view of (19), we have then lim N →∞ N u N = 0. This last convergence can be recast more explicitly as l p P S m > m 1 / 2 ( N ) = 0 , N i m N m su 1 which is clearly equivalent to (13). 4CharacterizingtheHo¨lderianFCLT Before proving our main result let us set assumptions for ρ ( h ). Definition 6. We denote by R the class of non decreasing functions ρ satisfying i) for some 0 < α 1 / 2, and some positive function L which is normalized slowly varying at infinity, ρ ( h ) = h α L (1 /h ) , 0 < h 1; (20) ii) θ ( t ) = t 1 / 2 ρ (1 /t ) is C 1 on [1 , ); iii) there is a β > 1 / 2 and some a > 1, such that θ ( t ) ln β ( t ) is non decreasing on [ a, ). Remark 7. Clearly L ( t ) ln β ( t ) is normalized slowly varying for any β > 0, so when α < 1 / 2, t 1 / 2 α L ( t ) ln β ( t ) is ultimately non decreasing and iii) is automatically satisfied. The assumption ii) of C 1 regularity for θ is not a real restriction, since the function ρ (1 /t ) being α -regularly varying at infinity is asymptoticaly equivalent to a C α -regularly varying function ρ ˜(1 /t )(see[1]).ThenthecorrespondingHo¨lderiannormsareequivalent. Put b := inf t 1 θ ( t ). Since by iii), θ ( t ) is ultimately increasing and lim t →∞ θ ( t ) = , we can define its generalized inverse ϕ on [ b, ) by ϕ ( u ) := sup { t 1; θ ( t ) u } . (21) With this definition, we have θ ( ϕ ( u )) = u for u b and ϕ ( θ ( t )) = t for t a . Theorem 8. Let ρ ∈ R . Then X 1 FCLT( B, ρ ) if and only if X 1 CLT( B ) and for every A > 0 , t →∞ k k ≥ ( t ) = 0 . (22) lim t P X 1 Corollary 9. Let ρ ∈ R with α < 1 / 2 in (20). Then X 1 FCLT( B, ρ ) if and only if X 1 CLT( B ) and lim t P k X 1 k ≥ θ ( t ) = 0 . (23) t →∞ 8
FCLTinHo¨lderspaces
Corollary 10. Let ρ ( h ) = h 1 / 2 ln β ( c/h ) with β > 1 / 2 . Then X 1 FCLT( B, ρ ) if and only if X 1 CLT( B ) and E exp d k X 1 k 1 < , for each d > 0 . (24) Corollary 11. Let ρ ∈ R and B = R . Then X 1 FCLT( R , ρ ) if and only if E X 1 = 0 and either (22) or (23) holds according to the case α = 1 / 2 or α < 1 / 2 . Remark 12. The requirement “for every A > 0” in (22) cannot be avoided in general. For instance let us choose B = R , X 1 symmetric such that P {| X 1 | ≥ u } = exp( u/c ), ( c > 0) and ρ ( h ) = h 1 / 2 ln(1 /h ), so θ ( t ) = ln t . Clearly (22) is satisfied only for A > c , so X 1 / FCLT( R , ρ ). Proof of Theorem 8. The necessity of “ X 1 CLT( B )” is obvious while that of (22) is contained in Theorem 5. For the converse part, Kuelbs FCLT gives us for any m 1 and 0 s 1 < ∙ ∙ ∙ < s m 1 ξ n sr ( s 1 ) , . . . , ξ s n r ( s m ) D W ( s 1 ) , . . . , W ( s m ) in the space B m . In particular, Condition i) of Theorem 3 is automatically fulfilled. So the remaining work is to check Condition (11). Write for simplicity t k = t kj = k 2 j , k = 0 , 1 , . . . , 2 j , j = 1 , 2 , . . . In view of (10), it is sufficient to prove that J →∞ li n m s u P n s J u p j ρ (21 j ) n 1 / 21 m k a < x 2 j k ξ n ( t k +1 ) ξ n ( t k ) k ≥ ε o = 0 . (25) lim p To this end, we bound the probability in the left hand side of (25) by P 1 + P 2 where P 1 := P n J j s u l p og n ρ (21 j ) n 1 / 21 m k a x 2 j k ξ n ( t k +1 ) ξ n ( t k ) k ≥ ε o and P 2 := P n j> s l u o p g n ρ (21 j ) n 1 / 21 m k a x 2 j k ξ n ( t k +1 ) ξ n ( t k ) k ≥ ε o . Here and throughout the paper, log denotes the logarithm with basis 2, while ln denotes the natural logarithm (log(2 x ) = x = ln(e x )). Estimation of P 2 . If j > log n , then t k +1 t k = 2 j < 1 /n and therefore with t k [ i/n, ( i + 1) /n ), either t k +1 is in ( i/n, ( i + 1) /n ] or belongs to ( i + 1) /n, ( i + 2) /n , where 1 i n 2 depends on k and j . In the first case we have k ξ n ( t k +1 ) ξ n ( t k ) k = k X i +1 k 2 j n 2 j n 1 m i a x n k X i k . If t k and t k +1 are in consecutive intervals, then k ξ n ( t k +1 ) ξ n ( t k ) k ≤ k ξ n ( t k ) ξ n (( i + 1) /n ) k + k ξ n (( i + 1) /n ) ξ n ( t k +1 ) k 2 j +1 n 1 m i a x k X i k . n 9
A.RaˇckauskasandCh.Suquet
With both cases taken into account we obtain P 2 P n j> s l u o p g n ρ (21 j ) n 1 / 2 n 2 j +11 m i a x n k X i k ≥ ε o 1 ε P n j> s l u o p g n θ (2 j ) 1 m i a x n k X i k ≥ 2 o P n 1 m i a x n k X i k ≥ ε 2 j> m l i og n n θ (2 j ) o n P n k X 1 k ≥ 2 εθ ( n ) o , for n a (see Definition 6.iii)). Hence, due to (22), for each ε > 0, lim n →∞ P 2 = 0. Estimation of P 1 . Let u k = [ nt k ]. Then u k nt k 1 + u k and 1 + u k u k +1 nt k +1 1 + u k +1 . Therefore k ξ n ( t k +1 ) ξ n ( t k ) k ≤ k ξ n ( t k +1 ) S u k +1 k + k S u k +1 S u k k + k S u k ξ n ( t k ) k . Since k S u k ξ n ( t k ) k ≤ k X 1+ u k k and k ξ n ( t k +1 ) S u k +1 k ≤ k X 1+ u k +1 k we obtain P 1 P 1 , 1 + P 1 , 2 , where P 1 , 1 := P n J j s u l p og n ρ (21 j ) n 1 / 21 m k a x 2 j k S u k +1 S u k k ≥ ε 2 o 1 1 2 P 1 , 2 := P n J j m a lo x g n ρ (2 j ) n / 1 m i a x n k X i k ≥ 4 ε o . In P 1 , 2 , the maximum over j is realized for j = log n , so P 1 , 2 = P n θ (1 n ) 1 m i a x n k X i k ≥ 4 ε o n P n k X 1 k ≥ ε 4 θ ( n ) o , which goes to zero by (22). To estimate P 1 , 1 , we use truncation arguments. For a positive δ , that will be precised later, define X i := X i 1 k X i k ≤ δθ ( n ) , X i 0 := X i E X i , e e e e e where 1 ( E ) denotes the indicator function of the event E . Let S u k and P 1 , 1 be the e expressions obtained by replacing X i with X i in S u k and P 1 , 1 . Similarly we define S 0 u k and P 1 0 , 1 by replacing X i with X 0 i and ε with ε/ 2. Due to (22), the control of P 1 , 1 reduces e to that of P 1 , 1 because P 1 , 1 P e 1 , 1 + P n 1 m i a x n k X i k > δθ ( n ) o P e 1 , 1 + n P k X 1 k > δθ ( n ) . e Now to deal with centered random variables, we shall prove that P 1 , 1 P 0 . It uffices 1 , 1 s to prove that for n and J large enough, the following holds 1 u k +1 4 . (26) J j s u l p og n ρ (2 j ) n 1 / 21 m k a x 2 j X k E X e i k < ε i =1+ u k As j log n , 1 u k +1 u k n 2 j + 1 2 n 2 j , 0 k < 2 j , (27) 10
¨ FCLT in Holder spaces
so it suffices to have 2 n 1 / 2 k E X e 1 k J j m a l x og n 2 j ε (28) ρ (2 j ) < 4 . Writing 2 j (2 j ) = 2 j/ 2 (2 j ) and recalling that θ is non decreasing on [ a, ), we see that for J log a , (28) reduces to 2 J/ 2 2 nθ 1 ( / 2 2 J ) k E X e 1 k < ε 4 . (29) Now, as E X 1 = 0, we get t = P k X 1 k > δθ ( u ) δθ k E X e 1 k ≤ Z δθ ( n ) P k X 1 k > t ) d Z n 0 ( u ) d u. By (22), there is an u 0 ( δ ) 1 such that for u u 0 ( δ ), u P k X 1 k > δθ ( u ) 1, whence e δ θ ( nn )+ δ Z n θ ( u 2 )d u, n u 0 ( δ ) . k E X 1 k ≤ − u As θ ( u ) u 1 / 2 = ρ (1 /u ) is non increasing, this last integral is dominated by n 1 / 2 θ ( n ) R n u 3 / 2 d u = 2 n 1 θ ( n ). Now plugging the estimate e 1 θ ) (30) k E X 1 k ≤ δn ( n into the left hand side of (29) gives 2 J 2 / 2 nθ 1 ( / 2 2 J ) k E X e 1 k ≤ 22 δ J θ 2( J 2 /J 2 ) θn ( 1 / n 2 ) 22 J δ. This concludes (26) provided δ/ 2 J < ε/ 8. Estimation of P 1 0 , 1 . Recalling (27), we have log n P 1 0 , 1 j = X J P n n 1 / 21 m k a x 2 j k S 0 u k +1 S 0 u k k ≥ 4 ερ (2 j ) o 0 lo X g n P n 1 m k a x 2 j k S u + k 1 +1 u k S ) 0 u 1 k / k 2 4 ε 2 θ (2 j ) o j = J ( u k l j o=g X Jnk 2= X j 1 P n ( k uS k 0 u + k 1 +1 u k S ) 0 u 1 k / k 2 4 ε 2 θ (2 j ) o (31) At this stage we use tail estimates related to ψ γ -Orlicz norms (see (38) and (39) in Sec-tion 5). By Talagrand’s inequality (Theorem 15 below), (27), Lemmas 16 and 17, we get for 1 < γ 2, S 0 u k +1 S 0 u k 2 K γ 2 clt ( X 1 ) + ( u k +1 u k ) 1 / 2 1 k X 0 1 k ψ γ ( u k +1 u k ) 1 / ψ γ K 0 1 + n 2 j 1 / 2 1 ln 1 δϕθ ( δnθ )( n ) , with a constant K 0 depending only on γ and of the distribution of X 1 . 11
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