fernique, theorem, mathematics, specifically, measure, theory, result, about, gaussian, measures, banach, spaces, extends, finite, dimensional, result, that, gaussian, random, variable, exponential, tails, result, proved, 1970, mathematician, xavier, fernique,. 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 EditLet X be a separable Banach space Let m 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 m defined on the Borel sets of R by ℓ m A m ℓ 1 A displaystyle ell ast mu A mu ell 1 A is a Gaussian measure a normal distribution with zero mean Then there exists a gt 0 such that X exp a x 2 d m x lt displaystyle int X exp alpha x 2 mathrm d mu x lt infty A fortiori m equivalently any X valued random variable G whose law is m has moments of all orders for all k 0 E G k X x k d m x lt displaystyle mathbb E G k int X x k mathrm d mu x lt infty References EditFernique Xavier 1970 Integrabilite des vecteurs gaussiens Comptes Rendus de l Academie des Sciences Serie A B 270 A1698 A1699 MR0266263Giuseppe 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 vte Retrieved from https en wikipedia org w index php title Fernique 27s theorem amp oldid 1075387396, wikipedia, wiki, book, books, library,