Symmetric spaces and sequences of independent random variables
Let X be a symmetric Banach space on a non-atomic probability space. Let also \((\xi_ n)\) be a sequence of i.i.d. random variables, \(\xi_ n\in X\), which satisfies the following condition: for some \(a,b,c>0\), \(1<q<2\) and all \(x\geq c\) \[ (1)\quad ax^{-q}\leq P[| \xi_ 1| \geq x]\leq bx^{-q}. \] It is announced that if X is perfect or reflexive, then the closed subspace of X generated by \((\xi_ n)\) is isomorphic to \(\ell_ q\). The question of necessity of (1) and the case \(X=L_ p\), \(0<p<1\) are also considered.
subspace generated by i.i.d. random variables
scientific article; zbMATH DE number 3969744