Zero-one laws for polynomials in Gaussian random variables: A simple proof
Zero-one laws for polynomials in Gaussian random variables have already been studied.(7) They are established here by very simple arguments: Fubini's theorem and the rotational invariance of centered Gaussian measures. The proof is built on the Polarization formula that has received much attention in Refs. 1 and 5. Our point of view derives from the deep work of Borell.(2) In a natural way, these results extend to finite-order Gaussian chaos processes.
Key WordsGaussian measures Gaussian chaos zero-one laws
Unable to display preview. Download preview PDF.
- 1.Arcones, A., and Giné, E. (1993). On decoupling, series expansions and tail behavior of chaos processes,J. Th. Prob. 6, 101–122.Google Scholar
- 2.Borell, C. (1978). Tail probabilities on Gauss space, Springer, Berlin,Lect. Notes in Math. 644, 73–82.Google Scholar
- 3.Borell, C. (1984). On polynomial chaos and integrability,Prob. Math. Statist. 3, 191–203.Google Scholar
- 4.Fernique, X. (1985). Gaussian random vectors and their reproducing kernel Hilbert spaces, Technical Report, University of Ottawa.Google Scholar
- 5.Kwapien, S. (1987). Decoupling inequalities for polynomial chaos,Ann. Prob. 15, 1062–1071.Google Scholar
- 6.Ledoux, M., and Talagrand, M. (1991).Probability in Banach Spaces, Springer, New York.Google Scholar
- 7.Rosinski, J., Samorodnitsky, G., and Taqqu, M. S. (1993). Zero-one laws for multilinear forms in Gaussian and other infinitely divisible random variables,J. Mult. Anal. 46, 61–82.Google Scholar