Another View of the CLT in Banach Spaces

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.