# Fernique's theorem

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 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 Giuseppe Da Prato and Jerzy Zabczyk, Stochastic equations in infinite dimension, Cambridge University Press, 1992. Theorem 2.7 This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it. Categories: Probability theoremsTheorems in measure theoryMathematical analysis stubs

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