Quadratic zero-one laws for Gaussian measures and the distribution of quadratic forms

de Acosta, Alejandro

doi:10.1090/S0002-9939-1976-0405524-5

Quadratic zero-one laws for Gaussian measures and the distribution of quadratic forms
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by Alejandro de Acosta

Proc. Amer. Math. Soc. 54 (1976), 319-325

DOI: https://doi.org/10.1090/S0002-9939-1976-0405524-5

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Erratum: Proc. Amer. Math. Soc. 61 (1976), 187.

Abstract:

Given a centered Gaussian measure on a vector space, the set of points where a sequence of quadratic forms converges to a constant has probability zero or one. The distribution of a quadratic form under a Gaussian measure is either point mass at zero or absolutely continuous.

References

Alejandro de Acosta, Stable measures and seminorms, Ann. Probability 3 (1975), no. 5, 865–875. MR 391202, DOI 10.1214/aop/1176996273
Albert Badrikian, Séminaire sur les fonctions aléatoires linéaires et les mesures cylindriques, Lecture Notes in Mathematics, Vol. 139, Springer-Verlag, Berlin-New York, 1970 (French). MR 0279271
Albert Badrikian and Simone Chevet, Mesures cylindriques, espaces de Wiener et fonctions aléatoires Gaussiennes, Lecture Notes in Mathematics, Vol. 379, Springer-Verlag, Berlin-New York, 1974 (French). MR 0420760
Kai Lai Chung, A course in probability theory, Harcourt, Brace & World, Inc., New York, 1968. MR 0229268
R. M. Dudley, On sequential convergence, Trans. Amer. Math. Soc. 112 (1964), 483–507. MR 175081, DOI 10.1090/S0002-9947-1964-0175081-6
R. M. Dudley, Sample functions of the Gaussian process, Ann. Probability 1 (1973), no. 1, 66–103. MR 346884, DOI 10.1214/aop/1176997026
R. M. Dudley and Marek Kanter, Zero-one laws for stable measures, Proc. Amer. Math. Soc. 45 (1974), 245–252. MR 370675, DOI 10.1090/S0002-9939-1974-0370675-9
Xavier Fernique, Régularité de processus gaussiens, Invent. Math. 12 (1971), 304–320 (French). MR 286166, DOI 10.1007/BF01403310
J. Kuelbs and T. Kurtz, Berry-Esseen estimates in Hilbert space and an application to the law of the iterated logarithm, Ann. Probability 2 (1974), 387–407. MR 362427, DOI 10.1214/aop/1176996655
Helmut H. Schaefer, Topological vector spaces, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1966. MR 0193469