Some asymptotic properties of the local time of the uniform empirical process

Abstract

We study the almost sure asymptotic properties of the local time of the uniform empirical process. In particular, we obtain two versions of the law of the iterated logarithm for the integral of the square of the local time. It is interesting to note that the corresponding problems for the Wiener process remain open. Properties of L^p-norms of the local time are studied. We also characterize the joint asymptotics of the local time at a fixed level and the maximum local time.