Stochastic basis in Fréchet space
The notion of a stochastic basis was introduced by \textit{W. Herer} [Demonstr. Math. 14, 719-724 (1981; Zbl 0513.46006)]. Let E be a Fréchet space and \(\mu\) be a Radon probability measure on E. Then a biorthogonal sequence \(\{x_ n,x'\!_ n\}\subset E\times E'\) is called a stochastic basis of (E,\(\mu)\) if \(\sum^{n}_{i=1}<x,x'\!_ i>x_ i\to x\) in E \(\mu\)-almost everywhere. It is known that for every Gaussian Radon measure \(\mu\), (E,\(\mu)\) has a stochastic basis and there exists a Banach subspace \(B\subset E\) with the Schauder basis such that \(\mu (B)=1\). In this paper, these results are generalized as follows. (1) Let \(\mu\) be a centered Radon probability measure on E of second order. If there exist sufficiently rich stochastically indenpendent continuous linear functionals, then (E,\(\mu)\) has a stochastic basis. (2) Let \(\mu\) be a Radon probability measure on E such that (E,\(\mu)\) has a stochastic basis. Then there exists a Banach subspace \(B\subset E\) with the Schauder basis such that \(\mu (B)=1\).
stochastically indenpendent continuous linear functionals
scientific article; zbMATH DE number 3946879