Fernique's theorem

Template:Short description 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

$\underset{X}{\int }\exp (\alpha \Vert x{\Vert }^2)\mathrm{d}\mu (x)<+\mathrm{\infty }.$

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

$𝔼[\Vert G{\Vert }^k]=\underset{X}{\int }\Vert x{\Vert }^k\mathrm{d}\mu (x)<+\mathrm{\infty }.$

References

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