Asymptotic normality of the truncation probability estimator for truncated dependent data

In some long term studies, a series of dependent and possibly truncated life-times may be observed. suppose that the lifetimes have a common marginaldistribution function. in left-truncation model, one observes data (xj,tj)only, when t'i < x{. under some regularity conditions, we provide a strongrepresentation of the j3n estimator of /? = /'( /, < xi), in the form of anaverage of random variables plus a remainder term. this representation en-ables us to obtain the asymptotic normality for j3n.