Characterization of Hilbert spaces by the strong law of large numbers (Q1084740)

Let X be a separable Banach space, \((\xi_ n)_{n\in N}\) be a sequence of independent strong second order mean zero random elements in X. It is shown that the convergence of the series \(\sum n^{-2}S_{\xi_ n}\) \((S_{\xi_ n}\) is the covariance operator of \(\xi_ n)\) in the space of nuclear operators implies \(n^{-1}\| \sum^{n}_{k=1}\xi_ k\| \to 0\) a.s. if and only if X is isomorphic to a Hilbert space.