SUDAKOVS TYPICAL MARGINALS, RANDOM LINEAR FUNCTIONALS AND A CONDITIONAL CENTRAL-LIMIT-THEOREM

V.N. Sudakov [Sud78] proved that the one-dimensional marginals of a hi gh-dimensional second order measure are close to each other in most di rections. Extending this and a related result in the context of projec tion pursuit of P. Diaconis and D. Freedman [Dia84], we give for a pro bability measure P and a random (a.s.) linear functional F on a Hilber t space simple sufficient conditions under which most of the one-dimen sional 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 theo rem (theorem 3) for random averages of triangular arrays of random var iables which satisfy only fairly weak asymptotic orthogonality conditi ons.