# A short proof of the levy continuity theorem in Hilbert space

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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 space*H*, certain functions*φ* have been shown by L. Gross to give rise to random variables*φ*∼ 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 functions*φ* of probability distributions*m* on*H*, given by*φ(y)*=∫*e* ^{i(y,x)}dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Let*φ* _{j} be the characteristic function of the probability measure*m* _{j} on*H*, Then necessary and sufficient that ∫*f dm* _{j} → ∫*f dm* for some probability measure*m* and all bounded continuous*f*, is that there exists a function*φ*, uniformly τ-continuous near zero, with*φ* _{j}∼ →*φ*∼ in probability.*φ* turns out, of course, to be the characteristic function of*m*. In the present paper we give a short proof of this theorem.

## Keywords

Hilbert Space Probability Measure Characteristic Function Short Proof Algebraic Operation## Preview

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## References

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