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Motivated by the central limit problem for convex bodies, we study normalapproximation of linear functionals of high-dimensional random vectors withvarious types of symmetries. In particular, we obtain results for distributionswhich are coordinatewise symmetric, uniform in a regular simplex, orspherically symmetric. Our proofs are based on Stein's method of exchangeablepairs; as far as we know, this approach has not previously been used in convexgeometry and we give a brief introduction to the classical method. Thespherically symmetric case is treated by a variation of Stein's method which isadapted for continuous symmetries.Comment: AMS-LaTeX, uses xy-pic, 23 pages; v3: added new corollary to Theorem

The Central Limit Problem for Random Vectors with Symmetries