Characterization of the cluster set of the LIL sequence in Banach space (Q1822818)

A characterization of the cluster set for the sequence \(\{S_ n(2 \log \log n)^{-1/2}\}\) where \(S_ n\) is the n-th sum of i.i.d. r.v.s \(\{X_ k\}\) taking values in a separable Banach space and having mean zero and finite second moment of the norm is obtained. The cluster set is either empty or has the form \(\alpha\) K where \(\alpha\in [0,1]\) and K is the unit ball of the reproducing kernel Hilbert space associated to the covariance of \(X_ 1\). The concrete form of the cluster set is determined by convergence or divergence of a special series.