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arxiv: math/0505618 · v3 · pith:BYDRAJ3Xnew · submitted 2005-05-27 · 🧮 math.PR · math.MG

The central limit problem for random vectors with symmetries

classification 🧮 math.PR math.MG
keywords methodsymmetricsymmetriescentralconvexlimitproblemrandom
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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 and we give a brief introduction to the classical method. The spherically symmetric case is treated by a variation of Stein's method which is adapted for continuous symmetries.

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