Fernique's theorem

Short description: A result about Gaussian measures on Banach spaces

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 Xavier Fernique.

Statement

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

[math]\displaystyle{ ( \ell_{\ast} \mu ) (A) = \mu ( \ell^{-1} (A) ), }[/math]

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

[math]\displaystyle{ \int_{X} \exp ( \alpha \| x \|^{2} ) \, \mathrm{d} \mu (x) \lt + \infty. }[/math]

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

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

References

• Fernique, Xavier (1970). "Intégrabilité des vecteurs gaussiens". Comptes Rendus de l'Académie des Sciences, Série A-B 270: A1698–A1699.  MR0266263

• Da Prato, Giuseppe; Zabczyk, Jerzy (1992). Stochastic equations in infinite dimension. Cambridge University Press. Theorem 2.7. ISBN 0-521-38529-6.