A short proof of the levy continuity theorem in Hilbert space

Abstract

A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ j be the characteristic function of the probability measurem j onH, Then necessary and sufficient that ∫f dm j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem.