On the occupation time of an iterated process having no local time

We study the asymptotic behaviour of the occupation time process ∫t0 IA(W1(L2(s)))ds, t ⩾ 0, where W1 is a standard Wiener process and L2 is a Wiener local time process at zero that is independent from W1. We prove limit laws, as well as almost sure upper and lower class theorems. Possible extensions of the obtained results are also discussed.