For a sequence of partial sums ofd-dimensional independent identically distributed random vectors a corresponding multivariate renewal process is defined componentwise. Via strong invariance together with an extreme value limit theorem for Rayleigh processes, a number of weak asymptotic results...

We prove the weak convergence of the empirical process of a(n) close pairs. The limit process is always Gaussian but its form depends on how fast the sequence a(n) goes to zero.