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A characterization of Gaussian law in Hilbert space. (English) Zbl 0733.60032

Let \({\mathcal H}\) be a real separable Hilbert space and let X and Y be independent random variables taking values in \({\mathcal H}\). The aim of the paper is to prove that \(X+Y\) and X-Y are independent if and only if each of X and Y is Gaussian. The proof is based on solving a functional equation satisfied by the characteristic functions of X and Y.

MSC:

60E07 Infinitely divisible distributions; stable distributions
39B52 Functional equations for functions with more general domains and/or ranges
46C15 Characterizations of Hilbert spaces
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References:

[1] Eaton, M. L. andPathak, P. K.,A characterization of the normal law on Hilbert space. Sankhya Ser. A 31 (1969), 259–268. · Zbl 0183.48608
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[3] Rao, C. R.,Some problems in the characterization of the multivariate normal distribution. (Inaugural Linnik Memorial Lecture). InStatistical distributions in scientific work, Vol. 3. D. Reidel Publishing Co., Dordrecht-Boston, 1975, pp. 1–13.
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