A strengthened Orlicz–Pettis theorem via Itô–Nisio

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