一类椭球等高矩阵分布的矩

ON THE MOMENTS OF A KIND OF ELLIPTICAL MATRIX DISTRIBUTIONS

摘要: 设X是m×n随机矩阵,n≥m,S=XX’,O_m是所有m×m正交阵的集合。如果对任意的Γ∈O_m,\Gamma X \stackreld= X 则对任意整数k E（S^k）=c^kI_m cov（vec S^k）=α_kI_m²+β_kK_m²+γ_kQ_m²其中 c_k、α_k、β_k、和γ_k是某些常数; I_i,l×l单位阵; \beginaligned K_m 2 & =\sum_i j=1^m H_i \otimes H_i j^\prime ; \\ Q_m^2 & =\sum_i j=1^m H_i j \otimes H_i j ;\endaligned;而 H_ij表示这样的m×m矩阵,除了h_ij=1外,其它元素为零, \otimes 表示 Kronecker积。另外,本文也求出了一些特殊的α_k,β_k,γ_k和c_k的值。

Abstract: Let X be an m×n random matrix, n≥m and S=XX’. If Γ∈O_m,\Gamma X \stackreld= X then for any integer k we get E（S^k）=c^kI_m cov（vec S^k）=α_kI_m²+β_kK_m²+γ_kQ_m² Where α_k,β_k,γ_kandc_k are some constants, \beginaligned K_m 2 & =\sum_i j=1^m H_i \otimes H_i j^\prime ; \\ Q_m^2 & =\sum_i j=1^m H_i j \otimes H_i j ;\endaligned here H_ij denotes the m×m matrix with h_ij=1 and all other elements zero and \otimes denotes Kroncker producte. Some special α_k,β_k,γ_kandc_k are found.