Banach random walk in the unit ball \(S\subset l^{2}\) and chaotic decomposition of \(l^{2}( S,\mathbb {P})\) (Q501840)

The paper combines Banach's concept of Lebesgue integral in abstract spaces and the expectation with respect to the measure \(\mathbb{P}\) induced by a Banach random walk. The random walk is considered in the unit ball \(S\) in the space \(l_2\). According to Banach's result, one of the very common nonnegative linear functionals defined on \(S\) is the limit of the integral functionals involving the sequence of density functions. The expression for these density functions can be easily reinterpreted in probabilistic terms. The expression of the sequence of pre-limit functionals in terms of the Banach random walk is given, and an orthogonal expansion of square integrable functionals of the Banach random walk in terms of Legendre polynomials is obtained. As a consequence, a decomposition of \(l_2(S,\mathbb{P})\) in terms of Banach chaoses is given.