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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
Article

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.

Keywords

Hilbert Space Probability Measure Characteristic Function Short Proof Algebraic Operation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

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Copyright information

© Hebrew University 1965

Authors and Affiliations

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

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