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Another View of the CLT in Banach Spaces

Author

Listed:
  • Jim Kuelbs

    (University of Wisconsin)

  • Joel Zinn

    (Texas A&M University)

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.

Suggested Citation

  • Jim Kuelbs & Joel Zinn, 2008. "Another View of the CLT in Banach Spaces," Journal of Theoretical Probability, Springer, vol. 21(4), pages 982-1029, December.
    • Handle: RePEc:spr:jotpro:v:21:y:2008:i:4:d:10.1007_s10959-008-0166-6
    DOI: 10.1007/s10959-008-0166-6
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    References listed on IDEAS

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    1. de Araujo, Aloisio Pessoa, 1978. "On the central limit theorem in Banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 598-613, December.
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