  A strong law of large numbers for $d$-dimensional random projections of the
$n$-dimensional cube is derived. It shows that with respect to the Hausdorff
distance a properly normalized random projection of $[-1,1]^n$ onto
$\mathbb{R}^d$ almost surely converges to a centered $d$-dimensional Euclidean
ball of radius $\sqrt{2/\pi}$, as $n\to\infty$. For every point inside this
ball we determine the asymptotic number of vertices and the volume of the part
of the cube projected `close' to this point. Moreover, large deviations for
random projections of general product measures are studied. Let $\nu^{\otimes
n}$ be the $n$-fold product measure of a Borel probability measure $\nu$ on
$\mathbb{R}$, and let $I$ be uniformly distributed on the Stiefel manifold of
orthogonal $d$-frames in $\mathbb{R}^n$. It is shown that the sequence of
random measures $\nu^{\otimes n}\circ(n^{-1/2}I^*)^{-1}$, $n\in\mathbb{N}$,
satisfies a large deviations principle with probability $1$. The rate function
is explicitly identified in terms of the moment generating function of $\nu$.
At the heart of the proofs lies a transition trick which allows to replace the
uniform projection by the Gaussian one. A number of concrete examples are
discussed as well, including the uniform distributions on the cube $[-1,1]^n$
and the discrete cube $\{-1,1\}^n$ as a special cases.
