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This paper addresses the question of when projections of a high-dimensionalrandom vector are approximately Gaussian. This problem has been studiedpreviously in the context of high-dimensional data analysis, where the focus ison low-dimensional projections of high-dimensional point clouds. The focus ofthis paper is on the typical behavior when the projections are generated by ani.i.d. Gaussian projection matrix. The main results are bounds on the deviationbetween the conditional distribution of the projections and a Gaussianapproximation, where the conditioning is on the projection matrix. The boundsare given in terms of the quadratic Wasserstein distance and relative entropyand are stated explicitly as a function of the number of projections andcertain key properties of the random vector. The proof uses Talagrand'stransportation inequality and a general integral-moment inequality for mutualinformation. Applications to random linear estimation and compressed sensingare discussed.

Conditional Central Limit Theorems for Gaussian Projections