Fernique's theorem

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.

tatement of the theorem

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 "ℓ" : "X" → R, the push-forward measure "ℓ"_∗"μ" on R defined by

:$(\; ell\_\{ast\}\; mu\; )\; (A)\; =\; mu\; (\; ell^\{-1\}\; (A)\; )\; mbox\{\; for\; \}\; A\; subseteq\; X$

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,

:

References

* cite journal
last = Fernique
first = Xavier
title = Intégrabilité des vecteurs gaussiens
journal = C. R. Acad. Sci. Paris Sér. A-B
volume = 270
year = 1970
pages = A1698–A1699 MathSciNet|id=0266263