A note on compact differentiability and theδ-method

Abstract

Letη _n ,n ∈ ℕ, be arbitrary functions defined on a probability space (ω,A,P) with values in a normed vector spaceB ₁ ,μ ∈ B ₁ andξ ₀ a separable random element inB ₁ such thatξ _n :=√n(η _n −μ) converges weakly toξ ₀ in the sense of Hoffmann-Jørgensen. Then with (B ₂, ∥·∥₂) being another normed vector space andφ:B ₁→B ₂ compactly differentiable atμ with derivateD _μ, the random variable$\parallel \sqrt n (\phi (n_n ) - \phi (\mu )) - D_\mu (\sqrt n (n_n - \mu ))\parallel 2*$ converges to 0P-stochastically where “*” denotes the measurable cover. We show that the classicalδ — method extends to the non-measurable case where in the proof we shall not make use of any representation theorems but only of a slight refinement of the usual characterisation of compact differentiability, due to the fact that we will not assume {ξ _n :n ∈ ℕ} being tight.