THE ASYMPTOTIC-BEHAVIOR OF LOCAL-TIMES AND OCCUPATION INTEGRALS OF THE N-PARAMETER WIENER PROCESS IN R(D)

Let L(x, T), x is-an-element-of R(d), T is-an-element-of R+(N), be the local time of the N-parameter Wiener process W taking values. in R(d). Even in the distribution valued cases d greater-than-or-equal-to 2 N, 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 of L(x, T) as absolute value of x --> 0 and/or T --> infinity and of related occupation integrals X(T)(f) = integral [0,T] f(W(s)) ds as T --> infinity. We obtain the rates of explosion in laws of the first order, i.e. normalized convergence laws for L(x, T) resp. X(T)(f), and of the second order, i.e. normalized convergence laws for L(x, T)-E(L(x, T)) resp. X(T)(f)-E(X(T)(f)).