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

6.3. Some preliminary results 97Proposition 6.2. For β ∈ B, j ∈ {1, 2, 3} and t ∈ [0, 1] d , we haveand for b ≥ D 0√n˜h j (β)log(˜h j (β)), we haveE f(Z2β,j (t) ) ≤ σ2 ‖K β ‖ 2 2n˜h j (β) ,{}P f [‖Z β,j ‖ ∞ ≥ b] ≤ D 1˜h j (β) exp − b2 n˜h j (β)exp2‖K β ‖ 2 2σ 2where ˜h j (β) = ∏ di=1 h i,j(β).Proposition 6.3. For p > 0, β ∈ B and j ∈ {1, 2, 3}, we havelimsupn→∞ f∈Σ(β,L)lim supn→∞supf∈Σ(β,L)⎧⎨⎩√ ⎫D 2 (β)b n˜h j (β) ⎬(log(˜h j (β))) 1/2 ⎭ ,E f[(‖ ˆf β,j − f‖ ∞ (M j (β)ψ n (β)) −1 ) p ]≤ 1 (6.10)E f[(‖ ˆf β,j − f‖ ∞ ψ −1n (β)) pI{‖ ˆfβ,j −f‖ ∞ ≥(1+ε(n))M j (β)ψ n (β)}]= 0, (6.11)where ε(n) satisfies ε(n) ≥ (log n) −1/4 , and I A denotes the indicator function of a set A.Proposition 6.1, respectively Proposition 6.2, can be obtained following the proof of Proposition3.1, respectively Lemma 3.4, of Chapter 3. The proof of Proposition 6.3 can bededuced from the proofs of Chapter 3 and we add some elements of proof in Section 6.8.Define for t ∈ [0, 1] d , j ∈ {1, 2} and a function f the bias term of ˆf β∗γ,jb β∗γ,j (t, f) = E f ( ˆf β∗γ,j (t)) − f(t),and the stochastic term of ˆf β∗γ,jZ β∗γ,j (t) = ˆf β∗γ,j (t) − E f ( ˆf β∗γ,j (t)) =σ √ n∫K β∗γ (t − u)dW u .The estimator ˆf β∗γ,j satisfies the two lemmas.Lemma 6.3. For β, γ ∈ B, we havesup ‖b β∗γ,2 (·, f) − b γ,2 (·, f)‖ ∞ ≤f∈Σ(β,L)When B = {γ, β} with γ < β we haveandsupf∈Σ(β,L)‖b β,2 (·, f)‖ ∞ ≤ ψ n(β)λ 2 (β).2β + 1sup ‖b β∗γ,1 (·, f) − b γ,3 (·, f)‖ ∞ ≤ sup ‖b β,1 (·, f)‖ ∞ ≤ ψ n(β)f∈Σ(β,L)f∈Σ(β,L)2β + 1sup ‖b β∗γ,1 (·, f) − b β,1 (·, f)‖ ∞ ≤f∈Σ(γ,L)supf∈Σ(γ,L)‖b γ,3 (·, f)‖ ∞ ≤ ψ n(γ)λ 3 (γ).2γ + 1