On uniform integrability and asymptotically risk-efficient sequential estimation

In the theory of asymptotically risk-efficient sequential estimation, one encounters uniform integrability of both positive and negative powers of stopping rules and moment convergence of randomly stopped statistics. We describe a simple approach to obtain these uniform integrability and moment convergence results, not only in the classical setting of i.i.d. observations but also for much more general stochastic sequences. We also use this approach to establish asymptotic risk efficiency of sequential estimators of means of stochastic sequences and to derive asymptotic approximations for the mean squared errors of ratio estimators.