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 {∥Xn∥:n⩾1} is sufficient, but not necessary, whereas uniform tightness of {〈Xn,y〉:n⩾1,∥y∥⩽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.