A strengthened Orlicz–Pettis theorem via Itô–Nisio

Abstract

In this note, we deduce a strengthening of the Orlicz–Pettis theorem from the Itô–Nisio theorem. The argument shows that given any series in a Banach space which is not summable (or more generally unconditionally summable), we can construct a (coarse-grained) subseries with the property that—under some appropriate notion of “almost all”—almost all further subseries thereof fail to be weakly summable. Moreover, a strengthening of the Itô–Nisio theorem by Hoffmann–Jørgensen allows us to replace ‘weakly summable’ with ‘tau-weakly summable’ for appropriate topologies tau weaker than the weak topology. A treatment of the Itô–Nisio theorem for admissible tau is given.