Sudakov's typical marginals, random linear functionals and a conditional central limit theorem
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V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a high-dimensional second order measure are close to each other in most directions. Extending this and a related result in the context of projection pursuit of P. Diaconis and D. Freedman [Dia84], we give for a probability measure \(P\) and a random (a.s.) linear functional \(F\) on a Hilbert space simple sufficient conditions under which most of the one-dimensional images of \(P\) under \(F\) are close to their canonical mixture which turns out to be almost a mixed normal distribution. Using the concept of approximate conditioning we deduce a conditional central limit theorem (theorem 3) for random averages of triangular arrays of random variables which satisfy only fairly weak asymptotic orthogonality conditions.
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