Fernique's theorem is a result in probability theory, particularly in the context of Gaussian processes and stochastic analysis. It deals with the continuity properties of stochastic processes, specifically the continuity of sample paths of certain classes of random functions.
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A Gaussian random variable in a Banach space taking values in a separable Banach space satisfies the displayed exponential-integrability bound for some positive . In particular every finite power of its norm has a finite expected value. The value of depends on the Gaussian measure; the theorem does not assert this for every positive .