A short proof of the levy continuity theorem in Hilbert space
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.
KeywordsHilbert Space Probability Measure Characteristic Function Short Proof Algebraic Operation
Unable to display preview. Download preview PDF.
- 2.L. Gross,Harmonic analysis in Hilbert space, Memoirs, American Mathematical Society, 46, 1963.Google Scholar
- 3.A. N. Kolomogorov, “A note to the papers of R. A. Minlos and V. Sazonov,”Teoriya Veroy atnostei i eyo Primenyeniya 4 (1959), 237–239;Theory of Probability and its Applications 4 221–223.Google Scholar
- 5.Yu. V. Prohorov,Convergence of random processes and limit theorems in probability theory, Theory of probability and its applications,1 (1956) English translation published by S. I. A. M., 157–214.Google Scholar