Fernique's theorem

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 centred 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

$( \ell_{\ast} \mu ) (A) = \mu ( \ell^{-1} (A) ),$

is a Gaussian measure (a normal distribution) with zero mean. Then there exists α > 0 such that

$\int_{X} \exp ( \alpha \| x \|^{2} ) \, \mathrm{d} \mu (x) < + \infty.$

A fortiori, μ (equivalently, any X-valued random variable G whose law is μ) has moments of all orders: for all k ≥ 0,

$\mathbb{E} [ \| G \|^{k} ] = \int_{X} \| x \|^{k} \, \mathrm{d} \mu (x) < + \infty.$

References

• 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