Unusual cluster sets for the LIL sequence in Banach space (Q909327)

Let \(S_ n=X_ 1+...+X_ n\) be a sum of i.i.d. Banach space valued random variables with mean 0 and weak second moment. Let K be the unit ball of the reproducing kernel Hilbert space associated to the covariance of \(X_ 1\). Let A be the cluster set of \(\{S_ n/(2n \log \log n)^{1/2}\}.\) The main result says that for each \(\alpha\in [0,1)\) there exists \(C_ 0\)-valued random variables such that \(A=\alpha K\) a.s. Under strong second moment conditions a necessary and sufficient condition for \(A=\Phi\) is given.