A short proof of the levy continuity theorem in Hilbert space |
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Authors: | J Feldman |
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Institution: | (1) University of California, Berkeley, California |
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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.
Research supported by National Science Foundation Grant GP-3977. |
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