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

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6.6. Proof of Theorem 6.3 109By Proposition 6.3, we have lim n→∞ sup f∈Σ(β,L) Q 4,n (β, β, f) = 0. To prove (6.38), it isenough to prove that, for all γ ∈ B such that γ > β,limsupn→∞ f∈Σ(β,L)Q 4,n (β, γ, f) = 0. (6.43)Let γ ∈ B, such that γ > β, and f ∈ Σ(β, L). Following the same reasoning as in theproof of (6.22) from (6.31) to (6.33), using Proposition 6.1, Lemma 6.2 and Lemma 6.3,we deduce that‖ ˆf γ,2 − f‖ ∞ I { ˆβani =γ} ≤(‖b β∗γ,2(·, f) − b γ,2 (·, f)‖ ∞ + ‖b β∗γ,2 (·, f) − b β,2 (·, f)‖ ∞+‖b β,2 (·, f)‖ ∞ + ‖Z γ,2 ‖ ∞ )I { ˆβani =γ}( )2ψn (β)λ 2 (β)≤+ η 2 (β) + ‖Z γ,2 ‖ ∞ + |ϕ 2 | I2β + 1{ ˆβani =γ} ,≤ (M 2 (β) + ‖Z γ,2 ‖ ∞ + |ϕ 2 |) I { ˆβani =γ}(6.44)where ϕ 2 = ˆf β∗γ,2 (x 0 ) − ˆf[β,2 (x 0 ) − E f ˆfβ∗γ,2 (x 0 ) − ˆf]β,2 (x 0 ) , with some x 0 ∈ [0, 1] d . UsingLemma 6.4 and Proposition 6.2, we have that ϕ 2 is a N (0, πn) 2 variable, with variance πn2satisfyingπn 2 ≤ 2‖K β‖ 2 2σ 2 (1 + o(1)).n˜h 1 (β)We haveE f[‖Zγ,2 ‖ 2p ∞ψ −2pn(β) ] [≤ (τ n,2 (γ)) 2p ψn−2p (β) + ψn −2p (β)E f ‖Zγ,2 ‖ 2p ∞I {‖Zγ,2 ‖ ∞ >τ n,2 (γ)}].Reasoning as in the proof of Lemma 6.5, we have thatlim (ψ n(β)) −2pn→∞Since γ > β, we deduce thatlimsupf∈Σ(β,L)supn→∞ f∈Σ(β,L)E f[‖Zγ,2 ‖ 2p ∞I {‖Zγ,2 ‖ ∞ >τ n,2 (γ)}]= 0.E f[‖Zγ,2 ‖ 2p∞ψ −2pn (β) ] = 0. (6.45)Moreover, the variable ϕ 3 satisfies the properties{P f [|ϕ 2 | > δ n ψ n (β)t] ≤ exp −D 13 t 2√ }log n , t ≥ 0, (6.46)limsupn→∞ f∈Σ(β,L)E f[|ϕ2 | 2p ψ −2pn (β) ] = 0. (6.47)The property (6.46) comes from the fact that ϕ 2 is a N (0, π 2 n) variable, and the property(6.47) can be proved as (6.45) using (6.46) and the proof of Lemma 6.5.
110 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classesThe relations (6.45) and (6.47) and Proposition 6.3 imply that{sup(E f ‖ ˆf γ,2 − f‖ 2p ∞ψn−2p (β)I { ˆβani =γ}f∈Σ(β,L)}) 1/2is bounded above by a positive constant for n large enough. Now to have (6.43), it isenough to prove thatlim sup P f[C 1,n ∩ { ˆβ]ani = γ} = 0. (6.48)n→∞ f∈Σ(β,L)Using the relations (6.44) and (6.46), and following the proof of (6.22) from (6.34) to(6.36), we deduce (6.48), which finishes the proof.6.7 Proof of Theorem 6.4•Proof of (6.8)By Proposition 6.3, we havelim supn→∞supf∈Σ(β,L)Then to have (6.8), it is enough to prove thatlimsupn→∞ f∈Σ(β,L){E f ‖ ˆf}β,1 − f‖ p ∞ψ −p (β) ≤ 1E f{‖ ˆf γ,3 − f‖ p ∞ψ −pn (β)I { ˜fani2 = ˆf γ,3 }{The event ˜fani2 = ˆf} {γ,3 satisfies ˜fani2 = ˆf}γ,3 ⊂ A 3,n ∩ A 4,n , whereA 3,n =The event A 3,n satisfies the lemma:{‖ ˆf β∗γ,1 − ˆf}γ,3 ‖ ∞ > η 3 (γ) ,{A 4,n = ‖ ˆf β∗γ,1 − ˆf β,1 ‖ ∞ < ψ }n(γ)λ 3 (γ)(1 + ρ n ) ,2γ + 1Lemma 6.9. We have for n large enoughsup P f [A 3,n ] ≤ D 14 (log n) − 1 −2γ+1 n p(β−γ)(2β+1)(2γ+1) .f∈Σ(β,L)n}= 0 (6.49)
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THÈSE DE DOCTORAT DE L’UNIVERSIT
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Table des matières 1Table des mati
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Table des matières 3.2 Un théorè
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1.1. Objet de la thèse 5blanc gaus
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1.1. Objet de la thèse 7calculées
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1.1. Objet de la thèse 9Un troisi
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1.2. Principaux résultats 11où β
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1.2. Principaux résultats 13le Cha
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1.2. Principaux résultats 15où K
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1.2. Principaux résultats 17de fon
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1.2. Principaux résultats 19L > 0
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Chapitre 2Asymptotically exact esti
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2.2. The main result and the estima
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2.2. The main result and the estima
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2.3. Proofs 27We study the stochast
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2.3. Proofs 29[for ε introduced in
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2.3. Proofs 31where P θ,X is the d
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2.3. Proofs 33as n → ∞ and usin
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2.3. Proofs 35Thus some algebra and
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2.3. Proofs 37Such choice of n is p
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2.4. Appendix of Chapter 2 39• Fi
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3.1. Introduction 41where ψ n = (
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3.2. The estimator and main result
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3.3. Upper bound 45Proof. The stoch
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3.4. Lower bound 473.4 Lower boundB
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3.4. Lower bound 49With these preli
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3.5. Appendix of Chapter 3 51• If
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3.5. Appendix of Chapter 3 53where
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Chapitre 4Asymptotically exact mini
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4.2. Main result 57with ˜β define
Page 62 and 63: 4.2. Main result 59(cf. Chapter 5 f
Page 64 and 65: 4.3. Upper bound 61Then we have( )
Page 66 and 67: 4.3. Upper bound 63where N Bj (u) i
Page 68 and 69: 4.4. Lower bound 65is a Gaussian ra
Page 70 and 71: 5.1. Cadre général de l’optimal
Page 72 and 73: 5.2. Application au problème d’a
Page 84 and 85: 5.3. Lien entre l’optimal recover
Page 90 and 91: 5.4. Etudes des constantes 87sente
Page 92 and 93: 6.1. Introduction 89(where ‖g‖
Page 94 and 95: 6.2. Main results 91and we denote b
Page 96 and 97: 6.2. Main results 936.2.4 Exact asy
Page 98 and 99: 6.2. Main results 952. These result
Page 100 and 101: 6.3. Some preliminary results 97Pro
Page 102 and 103: 6.4. Proof of Theorem 6.1 99Let 0 <
Page 104 and 105: 6.4. Proof of Theorem 6.1 101where
Page 106 and 107: 6.5. Proof of Theorem 6.2 103withA
Page 108 and 109: 6.5. Proof of Theorem 6.2 105The qu
Page 110 and 111: 6.6. Proof of Theorem 6.3 107andwhe
Page 114 and 115: 6.7. Proof of Theorem 6.4 111The pr
Page 116 and 117: 6.8. Proofs of the lemmas and propo
Page 122 and 123: 119Annexe.1 Résultats sur les proc
Page 124 and 125: Bibliographie 121BibliographieAdler
Page 126 and 127: Bibliographie 123Fuller, A. T. (196
Page 128 and 129: Bibliographie 125Micchelli, C. A. a
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THÈSE DE DOCTORAT DE L'UNIVERSITÉ PARIS 6 Spécialité ...

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