Let α([0,1]p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d−2)n=#{(k1,…,kp)∈[1,n]p;S1(k1)=⋯=Sp(kp)} run by the independent, symmetric, ℤd-valued random walks S1(n), …,Sp(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman–Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality \[\bigl({\mathbb{E}}I_{n_{1}+\cdots +n_{a}}^{m}\bigr)^{1/p}\le \sum_{\mathop{k_{1}+\cdots +k_{a}=m}\limits_{k_{1},\ldots,k_{a}\ge 0}}{\frac{m!}{k_{1}!\cdots k_{a}!}}\bigl({\mathbb{E}}I_{n_{1}}^{k_{1}}\bigr)^{1/p}\cdots \bigl({\mathbb{E}}I_{n_{a}}^{k_{a}}\bigr)^{1/p}\] in the case of random walks.