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

112 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classes•Proof of (6.9)The proof will be similar to the proof in the proof of (6.22). Let f ∈ Σ(γ, L). We fix(1 1/4.δ n =log n)We have{E f ‖ ˜f} {ani2 − f‖ p ∞ψn−p (γ) ≤(1 + δ n ) p (M 3 (γ)) p + E f ‖ ˜f}ani2 − f‖ p ∞ψn −p (γ)I C2,n ,{≤(1 + δ n ) p (M 3 (γ)) p + E f ‖ ˆf}β,1 − f‖ p ∞ψn −p (γ)I C2,n ∩A n{+E f ‖ ˆf}γ,3 − f‖ p ∞ψn−p (γ)I C2,n ∩A c ,nwhere{C 2,n = ‖ ˜f}ani2 − f‖ ∞ ψn−1 (γ) > (1 + δ n )M 3 (γ) ,and A n = (A 3,n ∩ A 4,n ) c . By Proposition 6.3, we havelimsupn→∞ f∈Σ(γ,L)The event A 4,n satisfies the lemmaE f{‖ ˆf γ,3 − f‖ p ∞ψ p n(γ)I C2,n ∩A c }= 0.Lemma 6.10. For n large enough, we have[ ]sup P f Ac D 154,n ≤f∈Σ(γ,L)˜h 1 (β) exp { −D 16 (log n) D 17n } √ }D 18exp{−D 19 (log n)D 17n D 18 ,with D 17 ∈ R.Lemma 6.10 implies thatlimsupn→∞ f∈Σ(γ,L)Then to have (6.9), it is enough to prove thatlimsupn→∞ f∈Σ(γ,L){E f ‖ ˆf}β,1 − f‖ p ∞ψn −p (γ)I C2,n ∩A c = 0.4,nwhere A 5,n = C 2,n ∩ A c 3,n ∩ A 4,n On A c 3,n ∩ A 4,n , we haveE f{‖ ˆf β,1 − f‖ p ∞ψ −pn (γ)I A5,n}= 0, (6.50)‖ ˆf β,1 − f‖ ∞ ≤‖ ˆf β,1 − ˆf β∗γ,1 ‖ ∞ + ‖ ˆf β∗γ,1 − ˆf γ,3 ‖ ∞ + ‖ ˆf γ,3 − f‖ ∞≤η 3 (γ) + ψ n(γ)λ 3 (γ)(1 + ρ n )+ ‖2γ + 1ˆf γ,3 − f‖ ∞ .Using Proposition 6.3, we deduce that, for n large enough,{sup E f ‖ ˆf}β,1 − f‖ 2p∞ψn −2p (γ)I A5,nf∈Σ(γ,L)