On convergence of random linear functionals

When is the pointwise limit of the characteristic functions of a sequence {Xn} of random elements taking values in a real separable Hilbert space H the characteristic function of a H valued random element? Uniform tightness of {[short parallel]Xn[short parallel]:n[greater-or-equal, slanted]1} is sufficient, but not necessary, whereas uniform tightness of {<Xn,y>:n[greater-or-equal, slanted]1,[short parallel]y[short parallel][less-than-or-equals, slant]1} is necessary, but not sufficient. The sufficiency statement extends to a separable reflexive Banach space, whereas the necessity statement extends to a Banach space with separable dual.