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

3.3. Upper bound 45Proof. The stochastic term is a Gaussian process on [0, 1] d . To prove this proposition, weuse a more general lemma about the supremum of a Gaussian process (Lemma 3.4 in theAppendix of Chapter 3). We haveP f[ψn−1‖Z n ‖ ∞ ≥ 2βC ] [0z= P f2β + 1]sup |ξ t | ≥ r 0 ,t∈[0,1] dwithandξ t =r 0 = 2βC 0zψ n√ nh1 · · · h dσ(2β + 1)1√h1 · · · h d∫[0,1] d K n (u, t) dW u .We will apply Lemma 3.4 (cf. Appendix of this chapter) to the process ξ t on the sets ∆belonging to{}d∏S = ∆ = ∆ i : ∆ i ∈ {[0, h i ), [h i , 1 − h i ], (1 − h i , 1]} .i=1Let ∆ ∈ S. The process ξ t on ∆ has the form(1u1 − tξ t = √ 1Q , . . . ,h1 · · · h d∫[0,1] u )d − t ddW u ,h d dwhere Q(u 1 , . . . , u d ) = K(u 1 , . . . , u d ) ∏ di=1 g i(u i ) and⎧⎨ 1 if ∆ i = [h i , 1 − h i ]g i (u i ) = 2I [0,1] if ∆ i = [0, h i )⎩2I [−1,0] if ∆ i = (1 − h i , 1].The function Q satisfies ‖Q‖ 2 2 = ∫ R d Q 2 = ‖K‖ 2 2. Moreover we have the following lemmawhich will be proved in the Appendix of Chapter 3.Lemma 3.2. There exists a constant D 2 > 0 such that, for all t ∈ [−1, 1] d∫h 1R d (Q(t + u) − Q(u)) 2 du ≤ D 2( d∑i=1|t i | min(1/2,β i)) 2. (3.8)The process ξ t satisfies the conditions of Lemma 3.4 and in particular satisfies condition(3.12) of that lemma with α i = min(1/2, β i ) in view of Lemma 3.2. We have by Lemma 3.3h =d∏i=1h i = C1/β 0L 1/β∗( ) 1/(2β+1) log n,nr 2 02‖K‖ 2 2= z2 log n2β + 1 .