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

4.3. Upper bound 61Then we have( )∣∣E f ̂fn (t)− f(t) ∣ ≤≤d∑∫j=1RK˜β(u)|f j (t j + uh j ) − f j (t j )|dud∑∫L j |h j | β jj=1≤ C adψ n˜L∫RK˜β(u)|u| β jduRK˜β(u)|u|˜βdu ∑ j∈ΛL 1− ˜β˜β+1j(1 + o(1)) = C adψ n(1 + o(1)).2˜β + 1The first line comes from the equality ∫ R K˜β(u)du = 1 and the second line follows from thefact that f ∈ Σ ad (β, L). The last line is a consequence of the equality ∫ R K˜β(u)|u|˜βdu =12˜β+1 and of the fact that |h j| β j= o(ψ n ) for j /∈ Λ.Proposition 4.2. We have for z > 1[]P ‖Z n ‖ ∞ ≥2˜β2˜β + 1 zC adψ n ≤ D 0 n − (z2 −1)2 ˜β+1 (log n)D 1,where D 0 is a positive constant and D 1 ∈ R.Proof. To prove this proposition, we follow a similar scheme as in the proof of Lemma3.4. We consider the Gaussian process ξ t defined for t = (t 1 , . . . , t d ) ∈ R d bywhere h(n) = ( ) 1log nnξ t = √ ∫h(n)R d2 ˜β+1 C 1/˜βadd∑j=1( )1 uj − t jK˜βdW uh j h j. We will study the quantity v2 = sup t∈[0,1] d E[ξ 2 t ] and thesemi-metric ρ on R d defined by ρ(s, t) = √ E[(ξ t − ξ s ) 2 ] for t, s ∈ R d . We have fort = (t 1 , . . . , t d ) ∈ R d∫ ( d∑E[ξt 2 ] =h(n)R d j=1∑=‖K˜β‖ 2 2˜L 1˜βj∈Λ( ) ) 21 uj − t jK˜βdu =h j h jL1˜β+1jd∑j=1(1 + o(1)) = ‖K˜β‖ 2 ˜β+1˜β2˜L (1 + o(1)),h(n)h‖K˜β‖ 2 2 + d(d − 1)h(n)jwith o(1) tending to 0 as n tends to ∞. The third equality follows from the fact thath(n)/h j = (log n) −D for β j ≠ ˜β. Then v 2 ≤ ‖K˜β‖ 2 2˜L˜β+1˜β (1+o(1)). For s = (s1 , . . . , s d ), t =