Fernique's theorem
= Fernique's theorem
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{title2=$\mathbb E e^{a\|X\|_B^2}<\infty\quad\text{for some }a>0$}
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= Fernique theorem
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{synonym}
A <Gaussian random variable in a Banach space> taking values in a <separable Banach space> satisfies the displayed exponential-integrability bound for some positive $a$. In particular every finite power of its <norm> has a finite <expected value>. The value of $a$ depends on the <Gaussian measure>; the theorem does not assert this for every positive $a$.