An extension of a central limit theorem for a mean of a random number of observations is given. A natural application occurs in the area of fixed-width confidence intervals. We provide an example which shows that the standard procedure does not preserve the intended coverage probability uniformly over nontrivial sets of distribution functions. The major weak convergence result is used to provide conditions for and a simple proof of such uniformity. The results are also shown to hold for $M$-estimates of location.
"On the Uniformity of Sequential Procedures." Ann. Statist. 5 (5) 1039 - 1046, September, 1977. https://doi.org/10.1214/aos/1176343958