On convergence of random linear functionals

When is the pointwise limit of the characteristic functions of a sequence {X-n} of random elements taking values in a real reparable Hilbert space H the characteristic function of a H valued random element? Uniform tightness of {parallel to X(n)parallel to : n >= 1} is sufficient, but not necessary, whereas uniform tightness of {< Xn, y >, : n >= 1, parallel to y parallel to <= 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. (c) 2005 Elsevier B.V. All rights reserved.