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Abstract

Let P be a probability distribution on q-dimensional space. Necessary and sufficient conditions are derived under which a random d-dimensional projection of P converges weakly to a fixed distribution Q as q tends to infinity, while d is an arbitrary fixed number. This complements a well-known result of Diaconis and Freedman (1984). Further we investigate d-dimensional projections of ^P, where ^P is the empirical distribution of a random sample from P of size n. We prove a conditional Central Limit Theorem for random projections of ^P - P given the data ^P, as q and n tend to infinity.