Subspaces of symmetric spaces, generated by independent random variables
Let (\(\Omega\),\({\mathcal A},P)\) be a probability space, M a mathematical expectation, and \(M^{\beta}\) a mathematical expectation with respect to the \(\sigma\)-algebra \(\beta\). If X, Y are random variables, then \(<X,Y>\) is defined to be M XY. A Banach space E of random variables on [0,1] is said to be symmetric if (i) \(| X| \leq | Y|\), \(Y\in E\) implies that \(X\in E\), and \(\| X\|_ E\leq \| Y\|_ E;\) (ii) if Y is equimeasurable distribution of \(| X|\) and \(Y\in E\), then \(X\in E\), and \(\| X\|_ E\leq \| Y\|_ E.\) If N is an even convex function on (-\(\infty,\infty)\), \(N(u)>0\) if \(u\neq 0\), \(N(0)=0\), then \(L^*_ N=\{X:\) \(\| X\|_{r^*_ N}<\infty \}\), where \(\| X\|_{r^*_ N}=\inf \{b>0:\quad M(N(b^{- 1}X))\leq 1\},\) and G is the closure of \(L_{\infty}\) in \(L^*_ N\), where \(N(u)=\exp (u^ 2)-1.\) In the main results of this paper the authors show that if \(\{X_ j,\quad j=1,2,...\}\) is a sequence of random variables on [0,1] such that \(\inf \| X_ j-MX_ j\|_{L_ 1}>0;\quad \sup \| X_ j- MX_ j\|_{L_{\infty}}<\infty,\) then (i) \(\overline{span}(X_ j)\subseteq E\) is isomorphic to \(\ell_ 2\) if and only if \(G\subseteq E;\) (ii) \(\overline{span}(X_ j)\) is complemented in E if and only if \(G\subseteq E\subseteq G'\), where G' is the dual of G. The results indicated have previously been proved for Rademacher systems in papers by other authors, including \textit{V. A. Rodin} and \textit{E. M. Semenov} [Funkts. Anal. Prilozh. 13, No.2, 91-92 (1979; Zbl 0409.46029)].
scientific article; zbMATH DE number 3931762
scientific article