实正交矩阵的子矩阵的幂的迹的渐进分布

Asymptotic Distributions of Traces of Powers of the Submatrices of Real Orthogonal Matrices

设随机矩阵U属于n阶实正交群O(n),O(n)的分布是单位Haar分布,[U]m表示U的m阶顺序主子矩阵,记Q=n/m～(1/n/m)[U]m.文献(Diaconis P,Shahshahani M.J Appl Probab,1994,A31:49-62.)通过计算TrUj的联合矩得出对固定的整数k,当n充分大时(TrU,TrU2,…,TrUk)渐进于正态分布.利用Jack函数和对称群的特征标的恒等式,推广这一结论到U的子矩阵情形,即证明了随机向量(TrQ,TrQ2,…,TrQk)当m→+∞时依分布收敛于正态分布.对特殊实正交矩阵群SO(n)也有类似的结论.

Let O（n） stand for the group of real orthogonal matrices of size n×n equipped with the unit Haar measure.Let random matrix U∈O（n）,[U]m be the top left m×m block of U and Q=Q=n/m～（1/n/m）[U]m.By computing the joint moments of TrUj,P.Diaconis and M.Shahshahani obtained that for any positive integer k,（TrU,TrU2,…,TrUk） is asymptotically normally distributed for enough large n.In this paper,by using Jack functions and identity of characters of symmetric group,we prove that the random vector（TrQ,TrQ2,…,TrQk） converges weakly to normal distribution when m→＋∞.Similar result holds true for the special real orthogonal matrices group SO（n）.