  We consider Random Walk in Random Scenery, denoted $X_n$, where the random
walk is symmetric on $Z^d$, with $d>4$, and the random field is made up of
i.i.d random variables with a stretched exponential tail decay, with exponent
$\alpha$ with $1<\alpha$. We present asymptotics for the probability, over both
randomness, that $\{X_n>n^{\beta}\}$ for $1/2<\beta<1$. To obtain such
asymptotics, we establish large deviations estimates for the the
self-intersection local times process.
