Uniform lower bounds for randomly stopped Banach space valued random sums (Q918569)

Let \(X_ 1,X_ 2,..\). be independent, Banach space (B, \(\| \cdot \|)\) valued random elements with \(S_ n=X_ 1+X_ 2+...+X_ n\). Let T be any, possibly randomized, stopping time based on \(X_ n\) and let \(\phi\) (\(\cdot)\) be any nondecreasing continuous function on \([0,\infty)\) with \(\phi (0)=0\) such that \(\phi (cx)=c^{\alpha}\phi (x)\) for some fixed \(\alpha >0\) and for all \(x\geq 0\), \(c\geq 2.\) The author proves that there exists a universal constant \(c^*_{\alpha}\) depending only on \(\alpha >0\) such that \[ E\max_{1\leq n\leq \tilde T}\phi (\| S_ n\|)\leq c^*_{\alpha}E\max_{1\leq n\leq T}\phi (\| S_ n\|) \] where \(\tilde T\) has the same marginal distribution as T and is independent of \(\{X_ n\}\). An upper bound for \(c^*_{\alpha}\) is also constructed.