Moderate deviations and law of the iterated logarithm for intersections of the ranges of random walks

Abstract

Let $S_1(n),…,S_p(n)$ be independent symmetric random walks in $\mathbb Z^d$. We establish moderate deviations and law of the iterated logarithm for the intersection of the ranges $$\#\{S_1[0,n]∩⋯∩S_p[0,n]\}$$ in the case $d=2, p≥2$ and the case $d=3, p=2.$