Fernique's theorem

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

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

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

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

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

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

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.