Distances between random orthogonal matrices and independent normals

Let Γ_n be an n × n Haar-invariant orthogonal matrix. Let Z_n be the p × q upper-left submatrix of Γ_n, where p = p_n and q = q_n are two positive integers. Let G_n be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between^√nZ_n and G_n in terms of the total variation distance, the Kullback-Leibler distance, the Hellinger distance, and the Euclidean distance. We prove that each of the first three distances goes to zero as long as pq/n goes to zero, and not so if (p, q) sits on the curve pq = σn, whereσ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq²/n goes to zero, and not so if (p, q) sitsonthecurvepq² = σn. A previous work by Jiang (2006) shows that the total variation distance goes to zero if both p/^√n and q/^√n go to zero, and it is not true provided p = c^√n and q = d^√n with c and d being constants. One of the above results confirms a conjecture that the total variation distance goes to zero as long as pq/n → 0 and the distance does not go to zero if pq = σn for some constant σ.