Israel Journal of Mathematics

, Volume 3, Issue 2, pp 99–103 | Cite as

A short proof of the levy continuity theorem in Hilbert space

  • J. Feldman


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.


Hilbert Space Probability Measure Characteristic Function Short Proof Algebraic Operation 
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Copyright information

© Hebrew University 1965

Authors and Affiliations

  • J. Feldman
    • 1
  1. 1.University of CaliforniaBerkeley

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