In mathematics — specifically, in measure theory — Fernique's theorem is a result about Gaussian measures on Banach spaces. It extends the finite-dimensional result that a Gaussian random variable has exponential tails. The result was proved in 1970 by the mathematician Xavier Fernique.
Statement of the theorem
Let (X, || ||) be a separable Banach space. Let μ be a centered Gaussian measure on X, i.e. a probability measure defined on the Borel sets of X such that, for every bounded linear functional ℓ : X → R, the push-forward measure ℓ∗μ defined on the Borel sets of R by
- Fernique, Xavier (1970). "Intégrabilité des vecteurs gaussiens". C. R. Acad. Sci. Paris Sér. A-B 270: A1698–A1699. MR 0266263
- Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimension, Cambridge University Press, 1992. Theorem 2.6
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