  This paper addresses the question of when projections of a high-dimensional
random vector are approximately Gaussian. This problem has been studied
previously in the context of high-dimensional data analysis, where the focus is
on low-dimensional projections of high-dimensional point clouds. The focus of
this paper is on the typical behavior when the projections are generated by an
i.i.d. Gaussian projection matrix. The main results are bounds on the deviation
between the conditional distribution of the projections and a Gaussian
approximation, where the conditioning is on the projection matrix. The bounds
are given in terms of the quadratic Wasserstein distance and relative entropy
and are stated explicitly as a function of the number of projections and
certain key properties of the random vector. The proof uses Talagrand's
transportation inequality and a general integral-moment inequality for mutual
information. Applications to random linear estimation and compressed sensing
are discussed.
