  We investigate random walks in independent, identically distributed random
sceneries under the assumption that the scenery variables satisfy Cramer's
condition. We prove moderate deviation principles in dimensions two and larger,
covering all those regimes where rate and speed do not depend on the actual
distribution of the scenery. In the case of dimension four and larger we even
obtain precise asymptotics for the annealed probability of a moderate
deviation, extending a classical central limit theorem of Kesten and Spitzer.
In dimension three and larger, an important ingredient in the proofs are new
concentration inequalities for self-intersection local times of random walks,
which are of independent interest, whilst in dimension two we use a recent
moderate deviation result for self-intersection local times, which is due to
Bass, Chen and Rosen.
