Sample behavior of some random series (Q1356622)

Let \(\{X_j\}^\infty_{j=1}\) be a sequence of i.i.d. real-valued random variables and \(V_1\), \(V_2, \dots\) be fixed vectors in Banach space \(E\). The scaled-sample property \((\{X^{(j)}/ \gamma_n\}^n_{j=1} \to k\) a.s. as \(n\to \infty\), where \(\gamma_n>0\) and \(X^{(j)}\) are i.i.d. with law \(\mu)\) is investigated for distributions \(\mu\) which are the laws of random series of the form \(\sum^\infty_{j=1} X_jV_j\). The main result establishes this convergence for a large collection of possible distributions \(\mu\) including some discrete examples.