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The Central Limit Problem for Random Vectors with Symmetries

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Abstract

Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries.

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Authors and Affiliations

  1. Department of Mathematics, Case Western Reserve University, Cleveland, OH, 44106, USA

    Elizabeth S. Meckes & Mark W. Meckes

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  1. Elizabeth S. Meckes
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  2. Mark W. Meckes
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Correspondence to Mark W. Meckes.

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This work was done while at Stanford University.

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Meckes, E.S., Meckes, M.W. The Central Limit Problem for Random Vectors with Symmetries. J Theor Probab 20, 697–720 (2007). https://doi.org/10.1007/s10959-007-0119-5

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  • DOI: https://doi.org/10.1007/s10959-007-0119-5

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