Banach Random Walk in the Unit Ball $$S\subset l^{2}$$ and Chaotic Decomposition of $$l^{2}\left( S,{{\mathbb {P}}}\right) $$l2S,P |
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Authors: | Tadeusz Banek |
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Institution: | 1.Faculty of Management,Lublin University of Technology,Lublin,Poland |
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Abstract: | A Banach random walk in the unit ball S in \(l^{2}\) is defined, and we show that the integral introduced by Banach (Theory of the integral. Warszawa-Lwów, 1937) can be expressed as the expectation with respect to the measure \({{\mathbb {P}}}\) induced by this walk. A decomposition \(l^{2}\left( S,{{\mathbb {P}}}\right) =\bigoplus _{i=0}^{\infty } {{\mathfrak {B}}}_{i}\) in terms of what we call Banach chaoses is given. |
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