On distribution of the norm for normal random elements in the space of continuous functions |
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Authors: | Ivan Matsak Anatolij Plichko |
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Institution: | 1. Taras Shevchenko National University of Kyiv, Academician Glushkov Ave. 2/6, 03127, Kyiv, Ukraine 2. Cracow University of Technology, ul. Warszawska 24, 31155, Krakow, Poland
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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 ε-renormed so that the family $ {{\mathcal{F}}_X} $ becomes uniformly Lipschitz in the new norm for any B-valued nondegenerate normal random element X. |
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