In a recent paper Barlow and Proschan noted that similar independence results appeared both in life table analysis and in fixed interval analysis. In this note we present a general asymptotic independence result of which these results are special cases. Also some further applications are given. In essence, our method may be described as follows: Assume that we are interested in $g$ quantities $q_1, \cdots, q_g$, each constructed from a sequence of independent random variables, and that these sequences are conditionally independent given their random lengths. Then by the completion of each sequence with independent random variables we obtain $g$ independent sequences. Under rather general assumptions we are able to deduce asymptotic properties of the original sequence from the corresponding properties of the completed sequences. In particular, we are often able to prove the asymptotic independence of the quantities $q_1, \cdots, q_g$.
"On a General Asymptotic Independence Result in Statistics." Ann. Probab. 10 (3) 831 - 837, August, 1982. https://doi.org/10.1214/aop/1176993793