Exact relations between T-k, the kth temporal moments of the nucleation flux, and the derivative of its Laplace transform (LT) are established and applied to available asymptotic expressions for the LT, generalizing earlier results by Shneidman and Weinberg [J. Chem. Phys. 95, 9148 (1991); 97, 3629 (1992)] on the time-lag T-0. For any kgreater than or equal to2 the moments T-k are expressed through simple algebraic combinations of T-0 and T-1. The two lower moments can thus be used to parametrize the time-dependent flux, with parametrization being substantially different from earlier, nonasymptotic approximations. The leading asymptotic dependences are expected to be applicable to both discrete and continuous versions of the nucleation equation, and to arbitrary sizes in the growth region. Higher-order corrections at the critical size are also obtained. (C) 2003 American Institute of Physics.