Long random walk excursions and local time

It is well known that the number of excursions of short, as well as long, duration of a Wiener process away from x that are completed by time t can be used to determine its local time process [eta](x,t). Let M(a,x,n) be the number of excursions of duration greater than a of a simple symmetric random walk Sk,k = 0, 1, ..., away from that are of duration greater than a in length and are completed by time n. We show here that if M (a,x,n) is suitably regulated by requiring that a = an be not too big, then it can also determine its local time process [xi](x,n) with appropriate rates of convergence.simple symmetric random walk long excursions local time Wiener process measure du voisinage