%0 Journal Article %T Self-Intersection Times for Random Walk, and Random Walk in Random Scenery in dimensions d>4 %A Amine Asselah %A Fabienne Castell %J Mathematics %D 2005 %I arXiv %X 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. %U http://arxiv.org/abs/math/0509721v2