Fernique's theorem explained

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

( \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} ) d \mu (x) < + inf ty.

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

E [ \| G \| ^{k} ] = \int _{X} \| x \| ^{k} d \mu (x) < + inf ty.

References

• Fernique. Xavier. Intégrabilité des vecteurs gaussiens. Comptes Rendus de l'Académie des Sciences, Série A-B. 270. 1970. A1698 - A1699.
• Book: Da Prato, Giuseppe . Giuseppe Da Prato

. Giuseppe Da Prato . Jerzy . Zabczyk . Jerzy Zabczyk . Stochastic equations in infinite dimension . Cambridge University Press . 1992 . Theorem 2.7 . 0-521-38529-6 .