A Chernoff and Savage theorem on the asymptotic normality of 2-sample linear rank statistics is here established for random sample sizes. The proof parallels that of Pyke and Shorack (1968), hereafter referred to as PS. A mild restriction on the underlying distributions is needed in the present situation. A result of Pyke (1968) on the weak convergence of the 1-sample empirical process for random sample sizes in the ordinary uniform metric is here extended to other metrics. This extension provides an essential step in the present proof and is also of separate interest. The results extend immediately to $c$-samples.
Ronald Pyke. Galen R. Shorack. "Weak Convergence and a Chernoff-Savage Theorem for Random Sample Sizes." Ann. Math. Statist. 39 (5) 1675 - 1685, October, 1968. https://doi.org/10.1214/aoms/1177698149