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We develop asymptotic approximation results that can be applied to sequentialestimation and inference problems, adaptive randomized controlled trials, andother statistical decision problems that involve multiple decision nodes withstructured and possibly endogenous information sets. Our results extend theclassic asymptotic representation theorem used extensively in efficiency boundtheory and local power analysis. In adaptive settings where the decision at onestage can affect the observation of variables in later stages, we show that alimiting data environment characterizes all limit distributions attainablethrough a joint choice of an adaptive design rule and statistics applied to theadaptively generated data, under local alternatives. We illustrate how thetheory can be applied to study the choice of adaptive rules and end-of-samplestatistical inference in batched (groupwise) sequential adaptive experiments.

Asymptotic Representations for Sequential Decisions, Adaptive Experiments, and Batched Bandits