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

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)