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

104 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesFrom Lemma 6.7, we know that P f (A 1,n (β ′ )) is at most of order(log n) − 12β ′ −+1 n pβ max + pβ′2β max +1 2β ′ +1 thus(τ n,1 (γ)) p ψn −p (β)P f (A 1,n (β ′ )) ≤ D 8 (log n) − 12β ′ + pγ+1 2γ+1 − pβ2β+1n − pβ max + pβ′2β max +1 2β ′ − pγ+1 2γ+1 + pβ2β+1.Since β ′ ≤ γ < β ≤ β max , we obtain thatandlimsupn→∞ f∈Σ(β,L)limsupn→∞ f∈Σ(β,L)Using Lemma 6.5, we deduce thatlimsupn→∞ f∈Σ(β,L)ψ −pn (β)‖b γ,1 (·, f)‖ p ∞P f (A 1,n (β ′ )) = 0 (6.26)(τ n,1 (γ)) p ψn −p (β)P f (A 1,n (β ′ )) = 0.E f{ψ−pn (β)‖Z γ,1 ‖ p ∞I A1,n (β ′ )}= 0. (6.27)From (6.26) and (6.27), we conclude that the quantity (6.23) tends to 0 as n → ∞.•Proof of (6.22)( 1/41Let δ n =log n)and β ∈ B. We haveR 2,n (β) ≤ (1 + δ n ) p +≤ (1 + δ n ) p +supf∈Σ(β,L)∑{E f ‖ ˜f (p) − f‖ p ∞ψ −p (β)I { ˆβ(p) ≥β}∩{‖ ˜f (p) −f‖ ∞ ψ −1supγ∈B,γ≥βf∈Σ(β,L)nQ 2,n (β, γ, f),n (β)>1+δ n }}where{Q 2,n (β, γ, f) = E f ‖ ˆf γ,1 − f‖ p ∞ψ −p (β)I { ˆβ(p) =γ}∩{‖ ˆf γ,1 −f‖ ∞ ψ −1nn (β)>1+δ n }}.By Proposition 6.3, we have lim n→∞ sup f∈Σ(β,L) Q 2,n (β, β, f) = 0. To prove (6.22), sincethe cardinal of B is finite and lim n→∞ δ n = 0, it is enough to prove, for γ ∈ B with γ > β,thatlim sup Q 2,n (β, γ, f) = 0. (6.28)n→∞ f∈Σ(β,L)Here we prove the result (6.28). Let γ ∈ B such that γ > β and f ∈ Σ(β, L). Ifˆβ (p) = γ, since γ > β, we have ‖ ˆf γ,1 − ˆf β,1 ‖ ∞ ≤ η 1 (β). Then,‖ ˆf(γ,1 − f‖ ∞ I { ˆβ(p) =γ} ≤ ‖ ˆf γ,1 − ˆf β,1 ‖ ∞ + ‖ ˆf)β,1 − f‖ ∞ I { ˆβ(p) =γ}≤(η 1 (β) + ‖ ˆf)β,1 − f‖ ∞ I { ˆβ(p) =γ} . (6.29)