Abstract. Let Γn be an n × n Haar-invariant orthogonal matrix. Let Zn be the p × q upper-left submatrix of Γn, where p = pn and q = qn are two positive integers. Let Gn be a p × q matrix whose pq entries are independent standard normals. In this paper we consider the distance between √ nZn and Gn 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, where σ is a constant. However, it is different for the Euclidean distance, which goes to zero provided pq 2 /n goes to zero, and not so if (p, q) sits on the curve pq 2 = σn. A previous work by Jiang [17] 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.

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We establish various bounds on the solutions to a Stein equation for Poisson approximation in Wasserstein distance with non-linear transportation costs. The proofs are a refinement of those in [Barbour and Xia (2006)] using the results in [Liu and Ma (2009)]. As a corollary, we obtain an estimate of Poisson approximation error measured in L 2 -Wasserstein distance.

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