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We demonstrate the asymptotic real second order freeness of Haar distributedorthogonal matrices and an independent ensemble of random matrices. Our mainresult states that if we have two independent ensembles of random matrices witha real second order limit distribution and one of them is invariant underconjugation by an orthogonal matrix, then the two ensembles are asymptoticallyreal second order free. This captures the known examples of asymptotic realsecond order freeness introduced by Redelmeier [R1, R2].

Real Second Order Freeness and Haar Orthogonal Matrices