107.
Let
B denote a separable Banach space with norm ‖⋅‖, and let
μ be a probability measure on
B for which linear functionals have mean zero and finite variance. Then there is a Hilbert space
H
μ
determined by the covariance of
μ such that
H
μ
⊆
B. Furthermore, for all
ε>0 and
x in the
B-norm closure of
H
μ
, there is a unique point,
T
ε
(
x), with minimum
H
μ
-norm in the
B-norm ball of radius
ε>0 and center
x. If
X is a random variable in
B with law
μ, then in a variety of settings we obtain the central limit theorem (CLT) for
T
ε
(
X) and certain modifications of such a quantity, even when
X itself fails the CLT. The motivation for the use of the mapping
T
ε
(⋅) comes from the large deviation rates for the Gaussian measure
γ determined by the covariance of
X whenever
γ exists. However, this is only motivation, and our results apply even when this Gaussian law fails to exist.
Research partially supported by NSA Grant H98230-06-1-0053.
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