On almost symmetric sequences in \(L_ p\) (Q917148)

The sequence \((X_ n)\) of random variables is said to be K-symmetric in \(L_ p\) if E \(| X_ n|^ p\) is finite for any \(n=1,2,..\). and \[ K^{-1}\| a_ 1X_ 1+...+a_ nX_ n\|_ p\leq \| a_ 1X_{b(1)}+...+a_ nX_{b(n)}\|_ p\leq K\| a_ 1X_ 1+...+a_ nX_ n\|_ p \] for every \(n=1,2,...\), \((a_ 1,...,a_ n)\in R^ n\) and any permutation (b(1),...,b(n)) of (1,...,n). The sequence \((X_ n)\) is almost symmetric in \(L_ p\) if for any \(a>0\) there exists an \(N=N(a)\) such that \((X_ n)_{n>N}\) is \((1+a)\)- symmetric in \(L_ p.\) A characterization of almost symmetric subsequences of \((X_ n)\) is given.