On distribution of the norm for normal random elements in the space of continuous functions. (English) Zbl 1272.60018
Summary: 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 \(\mathcal F_X=\{\operatorname{P}(\| X-z\|\leqslant r):z\in C(S)\}\), \(r\geqslant 0\), are uniformly Lipschitz and that every separable Banach space \(B\) can be \(\epsilon\)-renormed so that the family \(\mathcal F_X\) becomes uniformly Lipschitz in the new norm for any \(B\)-valued nondegenerate normal random element \(X\).
MSC:
| 60G15 | Gaussian processes |
| 60B11 | Probability theory on linear topological spaces |
| 46B20 | Geometry and structure of normed linear spaces |
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