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

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6.4. Proof of Theorem 6.1 101where the ξ k are i.i.d. N (0, 1) variables and vn 2 = n ‖fσ 2 k,β ‖ 2 2 = 2 log n c 2β+11(β)(1 − ε) 2 . Now,by the independence of the ξ i , we have() ()1M∑ dP k1M∑ dP kP i < 1/τ n ≥ P i < 1 ( 1 dP iP i < 1 ). (6.15)M dP 0 M dP 0 2τ n M dP 0 2τ nk=1k=1,k≠iThe probabilities above satisfy, since the ξ k are N (0, 1) variables,() (1M∑ dP kP i < 1 1M∑=P i exp { )}ξ k v n − v 2 1n
102 Sharp adaptive estimation in sup-norm for d-dimensional Hölder classes6.5 Proof of Theorem 6.2Since Theorem 6.1 is true, in particular for set B satisfying Condition (P ), to proveTheorem 6.2, it is enough to show that{lim sup sup sup E f ‖ ˜f}(p) − f‖ p ∞ψn−p (β) ≤ 1. (6.20)n→∞β∈B f∈Σ(β,L)Here we prove (6.20). Since the cardinal of B is finite, we are going to prove that for allβ ∈ B,lim sup ∆(β, n) ≤ 1,n→∞wherewithand{∆(β, n) = sup E f ‖ ˜f}(p) − f‖ p ∞ψn−p (β) ≤ R 1,n (β) + R 2,n (β),f∈Σ(β,L){R 1,n (β) = sup E f ‖ ˜f}(p) − f‖ p ∞ψn−p (β)I { ˆβ(p)
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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
Page 54 and 55: 3.5. Appendix of Chapter 3 51• If
Page 56 and 57: 3.5. Appendix of Chapter 3 53where
Page 58 and 59: Chapitre 4Asymptotically exact mini
Page 60 and 61: 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 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 112 and 113: 6.6. Proof of Theorem 6.3 109By Pro
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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