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.
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 is a Gaussian measure (a normal distribution) with zero mean.
Then there exists α > 0 such that A fortiori, μ (equivalently, any X-valued random variable G whose law is μ) has moments of all orders: for all k ≥ 0,