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In this paper, we show that the difference between the empirical estimator and the Conditional value-at-risk can be written as a simple partial sum + a residual term. Starting from this decomposition, we prove a central limit theorem and some almost sure results for the empirical estimator, for a large class of stationary sequences. We also construct a confidence interval with asymptotic level, and we study its coverage level through two different sets of simulation.

Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case