CROSSINGS AND OCCUPATION MEASURES FOR A CLASS OF SEMIMARTINGALES

We show that 1/root epsilon {integral(-infinity)(infinity) f(u)k(epsil on)N(tau)(X epsilon)(u) du - integral(0)(tau) f(X-t)a(t)dt} converges in law (as a continuous process) to c(psi) f(0)(tau) f(X-t)a(t) dB(t), where X-t = integral(0)(t) a(s) dW(s) + integral(0)(t) b(s) ds, with W a standard Brownian motion, a. and b regular and adapted processes, X-epsilon(t) = integral(-infinity)(infinity)(1/epsilon)psi((t - u)/eps ilon)X-u du, psi a smooth kernel, N-t(g)(u) the number of roots of the equation g(s) = u, s is an element of (0, t], k(epsilon) = root pi ep silon\2\parallel to(2), f a smooth function, B a standard Brownian mot ion independent of W and c(psi) a constant depending only on psi.