Total variation approximation of random orthogonal matrices by Gaussian matrices (Q2181628)

Consider a random orthogonal matrix distributed according to the Haar measure on the orthogonal group \(\mathcal{O}(n)\), and let \(W_n\) be the \(p_n\times q_n\) upper left-hand block of such a matrix, for sequences \((p_n)\) and \((q_n)\) such that \(p_nq_n=o(n)\) as \(n\rightarrow\infty\). Letting \(X_n\) be a \(p_n\times q_n\) matrix of IID standard Gaussian random variables, the author shows that the total variation distance between \(\sqrt{n}W_n\) and \(X_n\) converges to zero as \(n\rightarrow\infty\). This result builds on previous work by \textit{P. Diaconis} and \textit{D. Freedman} [Ann. Inst. Henri Poincaré, Probab. Stat. 23, Suppl., 397--423 (1987; Zbl 0619.60039)] and more recent work by \textit{T. Jiang} [Ann. Probab. 34, No. 4, 1497--1529 (2006; Zbl 1107.15018)].