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I wrote this paper a couple of years ago where we discuss this kind of topic. On page 6, you see a formula that comes from a paper from Acerbi available in Szego's book: $$\sigma^2(ES^{(N)}_\alpha(X)) \overset{N>>1}{=} \frac{1}{N(1-\alpha)^2} \int_0^{F^{-1}(1-\alpha)} dx \int_0^{F^{-1}(1-\alpha)} dy \{ \min( F(x), F(y) ) - F(x)F(y) \}$$ This should ...

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