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-Jorgensen. 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.