# Banach Random Walk in the Unit Ball $S\subset l^{2}$ and Chaotic Decomposition of $l^{2}\left( S,{{\mathbb {P}}}\right)$

Journal of Theoretical Probability, Dec 2016

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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Banach Random Walk in the Unit Ball $S\subset l^{2}$ and Chaotic Decomposition of $l^{2}\left( S,{{\mathbb {P}}}\right)$, Journal of Theoretical Probability, 2016, DOI: 10.1007/s10959-015-0620-1