  We study products of random isometries acting on Euclidean space. Building on
previous work of the second author, we prove a local limit theorem for balls of
shrinking radius with exponential speed under the assumption that a Markov
operator associated to the rotation component of the isometries has spectral
gap. We also prove that certain self-similar measures are absolutely continuous
with smooth densities. These families of self-similar measures give higher
dimensional analogues of Bernoulli convolutions on which absolute continuity
can be established for contraction ratios in an open set.
