Self-Intersection Times for Random Walk, and Random Walk in Random Scenery in dimensions d>4

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 $1\mathrm{/}2<\beta <1$. To obtain such asymptotics, we establish large deviations estimates for the the self-intersection local times process.