Another View of the CLT in Banach Spaces |
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Authors: | Jim Kuelbs Joel Zinn |
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Affiliation: | (1) Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA;(2) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA |
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Abstract: | 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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Keywords: | Central limit theorem Banach space Best approximations Sub-Gaussian |
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