  In this paper, explicit error bounds are derived in the approximation of rank
$k$ projections of certain $n$-dimensional random vectors by standard
$k$-dimensional Gaussian random vectors. The bounds are given in terms of $k$,
$n$, and a basis of the $k$-dimensional space onto which we project. The random
vectors considered are two generalizations of the case of a vector with
independent, identically distributed components. In the first case, the random
vector has components which are independent but need not have the same
distribution. The second case deals with finite exchangeable sequences of
random variables.
