On distribution of the norm for normal random elements in the space of continuous functions

Abstract

We consider distributions of norms for normal random elements X in separable Banach spaces, in particular, in the space C(S) of continuous functions on a compact space S. We prove that, under some nondegeneracy condition, the functions \( {{\mathcal{F}}_X}=\left\{ {\mathrm{P}\left( {\left\| {X-z} \right\|\leqslant r} \right):\;z\in C(S)} \right\},\;r\geqslant 0 \), are uniformly Lipschitz and that every separable Banach space B can be e-renormed so that the family \( {{\mathcal{F}}_X} \) becomes uniformly Lipschitz in the new norm for any B-valued nondegenerate normal random element X.