## 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 [2] Parthasarthy, K. R.,Probability measure on metric spaces. Academic Press, New York, 1967. [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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