The asymptotic behaviour of local times and occupation integrals of theN-parameter Wiener process in R d

Abstract

LetL(x, T),x∈Rd,T∈R+N, be the local time of theN-parameter Wiener processW taking values inRd. Even in the distribution valued casedd≧2N,L can be described in a series representation by means of multiple Wiener-Ito integrals. This setting proves to be a good starting point for the investigation of the asymptotic behaviour ofL(x, T) as |x|→0 and/orT∞ and of related occupation integrals\(X_T (f) = \int\limits_{[0,T]} f (W_S )\) asT→∞. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws forL(x, T) resp.XT(f), and of the second order, i.e. normalized convergence laws forL(x, T)−E(L(x, T)) resp.XT(f)−E(XT(f)).