A complete proof of above lemma can be found on page 241 (Lemma 7.27) of

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Hint: 2\(cov(X_1, X_2) = E[(X_1-\tilde{X_1})(X_2-\tilde{X_2})]\),

where \( (\tilde{X_1}, \tilde{X_2}) \) is an independent copy with the same joint distribution function as \( (X_1, X_2) \).

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