Stochastic basis in Fréchet space
type: Item
instance of: scholarly article
title: Stochastic basis in Fréchet space
zbMATH Open document ID: 0589.46004
DOI: 10.1007/BF01457222
author: Yoshiaki Okazaki
published in: Mathematische Annalen
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164105
review text: 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 \(\mu (B)=1\).
Mathematics Subject Classification ID: 46A35
Mathematics Subject Classification ID: 46G12
Mathematics Subject Classification ID: 46A04
Mathematics Subject Classification ID: 28C20
zbMATH DE Number: 3946879
zbMATH Keywords: Banach support
zbMATH Keywords: stochastic basis
zbMATH Keywords: Fréchet space
zbMATH Keywords: Radon probability
zbMATH Keywords: biorthogonal sequence
zbMATH Keywords: Gaussian Radon measure
zbMATH Keywords: Schauder basis
zbMATH Keywords: stochastically indenpendent continuous linear functionals
MaRDI profile type: Publication
cites work: STOCHASTIC BASES IN FRECHÉT SPACES
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OpenAlex ID: W2000936295
description: scientific article; zbMATH DE number 3946879