  Let $P$ be a probability distribution on $q$-dimensional space. The so-called
Diaconis-Freedman effect means that for a fixed dimension $d << q$, most
$d$-dimensional projections of $P$ look like a scale mixture of spherically
symmetric Gaussian distributions. The present paper provides necessary and
sufficient conditions for this phenomenon in a suitable asymptotic framework
with increasing dimension $q$. It turns out, that the conditions formulated by
Diaconis and Freedman (1984) are not only sufficient but necessary as well.
Moreover, letting $\hat{P}$ be the empirical distribution of $n$ independent
random vectors with distribution $P$, we investigate the behavior of the
empirical process $\sqrt{n}(\hat{P} - P)$ under random projections, conditional
on $\hat{P}$.
