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Conditonal convergence implies convergence?

Note : All measures below are probability measures. Let $\mu_n(X,Y)$ be a random probability measure on $\mathbb C$ depending on two random variables X and Y with values in $\mathbb{R}^N$. Actually,...
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Total variation convergence of random matrices and convergence of empirical spectral distributions

In the paper https://arxiv.org/pdf/1411.5713.pdf, on page 17, the authors prove in Theorem 7 that the total variation distance between the joint distribution of the entries of certain Wishart matrices ...
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