Another View of the CLT in Banach Spaces

Jim Kuelbs and Joel Zinn ()
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Jim Kuelbs: University of Wisconsin
Joel Zinn: Texas A&M University

Journal of Theoretical Probability, 2008, vol. 21, issue 4, 982-1029

Abstract: 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.

Keywords: Central limit theorem; Banach space; Best approximations; Sub-Gaussian; 60F05; 60F17 (search for similar items in EconPapers)
Date: 2008
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DOI: 10.1007/s10959-008-0166-6

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