How Many Entries of A Typical Orthogonal Matrix Can Be Approximated By Independent Normals?
Abstract
We solve an open problem of Diaconis that asks what are the largest orders of p_n and q_n such that Z_n, the p_n\times q_n upper left block of a random matrix \bold{\Gamma}_n which is uniformly distributed on the orthogonal group O(n), can be approximated by independent standard normals? This problem is solved by two different approximation methods. First, we show that the {\it variation distance} between the joint distribution of entries of Z_n and that of p_nq_n independent standard normals goes to zero provided p_n=o(\sqrt{n}) and q_n=o(\sqrt{n}). We also show that the above variation distance does not go to zero if p_n=[x\sqrt{n}] and q_n=[y\sqrt{n}] for any positive numbers x and y. This says that the largest orders of p_n and q_n are o(n^{1/2}) in the sense of the above approximation. Second, suppose \bold{\Gamma}_n=(\gamma_{ij})_{n\times n} is generated by performing the Gram-Schmidt algorithm on the columns of \bold{Y}_n=(y_{ij})_{n\times n} where \{y_{ij}; 1\leq i, j \leq n\} are i.i.d. standard normals. We show that \epsilon_n(m):=\max_{1\leq i \leq n, 1\leq j \leq m}|\sqrt{n}\gamma_{ij}-y_{ij}| goes to zero in probability as long as m=m_n=o(n/\log n). We also prove that \epsilon_n(m_n)\to 2\sqrt{\alpha} in probability when m_n=[n\alpha/\log n] for any \alpha>0. This says that m_n=o(n/\log n) is the largest order such that the entries of the first m_n columns of \bold{\Gamma}_n can be approximated simultaneously by independent standard normals.Keywords
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