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Martingale central-limit theorems for pivotal sampling

Ordered pivotal sampling is one of the simplest algorithm to perform without-replacement unequal probability sampling. It has found uses in the context of longitudinal surveys and spatial sampling, and enables in particular a good spatial balance of the selected units. In this work, we follow the approach proposed by Ohlsson~(1986), and apply a martingale central-limit theorem to prove the asymptotic normality of the Horvitz-Thompson estimator under a design-based approach, and under a model-assisted approach. In particular, our model assumptions allow for correlations between values, which is of particular interest for applications in spatial sampling.