How many entries of a typical orthogonal matrix can be approximated by independent normals?

Jiang, Tiefeng

Ann. Probab., Tome 34 (2006) no. 1, p. 1497-1529 / Harvested from Project Euclid

Résumé

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×q_n upper left block of a random matrix Γ_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 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 Γ_n=(γ_ij)_n×n is generated by performing the Gram–Schmidt algorithm on the columns of Y_n=(y_ij)_n×n, where {y_ij;1≤i,j≤n} are i.i.d. standard normals. We show that $\varepsilon _{n}(m):=\max_{1\leq i\leq n,1\leq j\leq m}|\sqrt{n}\gamma_{ij}-y_{ij}|$ goes to zero in probability as long as m=m_n=o(n/logn). We also prove that $\varepsilon _{n}(m_{n})\to 2\sqrt{\alpha}$ in probability when m_n=[nα/logn] for any α>0. This says that m_n=o(n/logn) is the largest order such that the entries of the first m_n columns of Γ_n can be approximated simultaneously by independent standard normals.

Détail

Publié le : 2006-07-14
Classification:  Haar measure,  Gram–Schmidt algorithm,  large deviation,  maxima,  product distribution,  random matrix theory,  variation distance,  15A52,  60B10,  60B15,  60F05,  60F99,  62H10