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 (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 â : X â R, the push-forward measure âÂÂ<sub>âÂÂ</sub>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 âÂÂ¥ 0,