H. Vonweizsacker, SUDAKOVS TYPICAL MARGINALS, RANDOM LINEAR FUNCTIONALS AND A CONDITIONAL CENTRAL-LIMIT-THEOREM, Probability theory and related fields, 107(3), 1997, pp. 313-324
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.