Using the model of random censorship, a necessary and sufficient condition for the consistency of the standard (actuarial) life table estimate of a survival distribution is derived. We establish the asymptotic normality of this estimate, showing that Greenwood's variance formula is nearly correct. In the case of a continuous survival distribution we establish limiting normality for the product limit estimate and for the closely related cumulative hazard process. Some applications of these results are outlined.
"A Large Sample Study of the Life Table and Product Limit Estimates Under Random Censorship." Ann. Statist. 2 (3) 437 - 453, May, 1974. https://doi.org/10.1214/aos/1176342705