On the supports of stochastic processes of multiplicity one (Q2734973)

Let \({\mathbf X}= (X_n)_{n\in\mathbb{Z}}\) be a doubly infinite sequence of complex-valued square integrable centered random variables on a probability space \((\Omega,{\mathcal F},P)\), where \({\mathcal F}\) is the \(\sigma\)-algebra generated by \({\mathbf X}\). Assume that \({\mathbf X}\) has multiplicity one, that is, the closed linear span of \(X_n\), \(n\in\mathbb{Z}\), coincides with the collection of complex-valued square integrable centered functions on \(\Omega\). The authors study the support of the measure \(m\in P{\mathbf X}^{-1}\). The results obtained, too involved to be described here, have some impact on Banach's problem of existence of a transformation with simple Lebesgue spectrum.NEWLINENEWLINEFor the entire collection see [Zbl 0961.60001].