THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

6.8. Proofs of the lemmas and propositions 115∣∣K γ,2 ∗ f(x) − K γ,2 ∗ f(y) ∣ ∫=∣ K γ (u) {f(x 1 − u 1 h 1,2 (γ), . . . , x d − u d h d,2 (γ)) − f(y 1 − u 1 h 1,2 (γ), . . . , y d − u d h d,2 (γ))} du∣(∫ ) d∑ d∑≤ K γ (u)du L i |x i − y i | β i= L i |x i − y i | β ii=1i=1Then, K γ,2 ∗ f belongs to Σ(β, L). As a consequence, we have‖b β∗γ,2 (·, f) − b γ,2 (·, f)‖ ∞ = ‖b β,2 (·, K γ,2 ∗ f)‖ ∞ ≤Thus, from Proposition 6.1, we deduce thatsupf∈Σ(β,L)‖b β∗γ,2 (·, f) − b γ,2 (·, f)‖ ∞ ≤ ψ n(β)λ 2 (β).2β + 1The two other results can be proved exactly in the same way.sup ‖b β,2 (·, f)‖ ∞ .f∈Σ(β,L)Proof of Lemma 6.4We prove here the result for j = 2. The result for j = 1 can be proved exactly in thesame way. Let β = (β 1 , . . . , β d ), γ ∈ B such that γ < β and t ∈ [0, 1] d . We haveE f[Z2β∗γ,2 (t) ] =σ 2n˜h 2 2(β)˜h 2 2(γ)∫ (∫( ) (t − u − vK β K γh 2 (β)vh 2 (γ))dv) 2du,where the notation u, for two vectors u = (u t 1, . . . , u d ) and t = (t 1 , . . . , t d ), represents thevector (u 1 /t 1 , . . . , u d /t d ). By the generalized Minkowskii inequality (cf. Annex .3), wededuce that[E f Z2β∗γ,2 (t) ] ( ∫ (∫ ( ) ( ) )σ 21/2 2t − u − v v≤K 2n˜h 2 2(β)˜h 2 βKγdu) 2 dv2(γ)h 2 (β) h 2 (γ)(∫ ( ) )≤ σ2 ‖K β ‖ 2 22vKn˜h 2 2(β)˜h 2 γ dv = σ2 ‖K β ‖ 2 22(γ) h 2 (γ) n˜h 2 (β) .Now the proof of (6.12) is similar to the proof of Proposition 6.3, and then tothat of Lemma 3.4 of Chapter 3. The process Z β∗γ satisfies as Z β , E f[Z2β∗γ (t) ] ≤σ 2 ‖K β ‖ 2 2n˜h 2 (β) .Furthermore, to apply that argument, we need to bound the quantity[ (Z ] 2E f β∗γ (t) − Zβ∗γ 2 (s)) 2 1 ∣ ∣by a multiple of for s = (s 1 , . . . , s d ),∑ dn˜h 2 (β) i=1 ∣ s i−t ih i,2 (β)∣ β i