  Let $(Z_n)_{n\in\N}$ be a $d$-dimensional {\it random walk in random
scenery}, i.e.,
  $Z_n=\sum_{k=0}^{n-1}Y(S_k)$ with $(S_k)_{k\in\N_0}$ a random walk in $\Z^d$
and $(Y(z))_{z\in\Z^d}$ an i.i.d. scenery, independent of the walk. The
walker's steps have mean zero and finite variance. We identify the speed and
the rate of the logarithmic decay of $\P(\frac 1n Z_n>b_n)$ for various choices
of sequences $(b_n)_n$ in $[1,\infty)$. Depending on $(b_n)_n$ and the upper
tails of the scenery, we identify different regimes for the speed of decay and
different variational formulas for the rate functions. In contrast to recent
work \cite{AC02} by A. Asselah and F. Castell, we consider sceneries {\it
unbounded} to infinity. It turns out that there are interesting connections to
large deviation properties of self-intersections of the walk, which have been
studied recently by X. Chen \cite{C03}.
