\(\ell ^ n_ p\) superspaces of spans of independent random variables (Q1111224)

The following problem, introduced by \textit{A. Pelczynski} and \textit{H. P. Rosenthal} [Studia Math. 52, 263-289 (1975; Zbl 0297.46023)] is considered: Given a subspace X of \(L_ p\), dim X\(=n\), and \(\epsilon >0\), estimate the smallest \(m=m_ p(X,\epsilon)\) such that there is a subspace Y of \(L_ p\) with \(X\subset Y\) and \(d(Y,\ell^ m_ p)<1+\epsilon\). In particular, estimate \(m_ p=\sup (m_ p(n,\epsilon):\) dim X\(=n).\) The author proves that if \(\epsilon >0\) is small enough, and X is the subspace of \(L_ p\) spanned by n independent Rademacher functions or by n independent Gaussian random variables, then \(m_ p(X,\epsilon)\geq r^ n\) where \(r=r(\epsilon,p)>1.\) There is a further interesting result: Let \(1\leq p<2\). For every \(\epsilon >0\) there exist \(\delta =\delta (\epsilon,p)>0\) and \(r=r(\epsilon,p)>1\) such that for every \(n\in N\) there is an n- dimensional subspace \(X_ n\subset L_ p(0,1)\) such that \(d(X_ n,\ell^ n_ p)\leq 1+\epsilon\) and \(m_ p(X_ n,\epsilon)\geq r^ n\).