具有有限秩相关性的高维随机向量的样本典型相关系数
概率论
2022-06-14 v3
摘要
考虑两个随机向量 x ~ ∈ R p \widetilde{\mathbf x} \in \mathbb R^p x ∈ R p y ~ ∈ R q \widetilde{\mathbf y} \in \mathbb R^q y ∈ R q x ~ = A z + C 1 1 / 2 x \widetilde{\mathbf x}=A\mathbf z+\mathbf C_1^{1/2}\mathbf x x = A z + C 1 1/2 x y ~ = B z + C 2 1 / 2 y \widetilde{\mathbf y}=B\mathbf z+\mathbf C_2^{1/2}\mathbf y y = B z + C 2 1/2 y x ∈ R p \mathbf x\in \mathbb R^p x ∈ R p y ∈ R q \mathbf y\in \mathbb R^q y ∈ R q z ∈ R r \mathbf z\in \mathbb R^r z ∈ R r C 1 \mathbf C_1 C 1 C 2 \mathbf C_2 C 2 p × p p \times p p × p q × q q\times q q × q A A A B B B p × r p\times r p × r q × r q\times r q × r ( x ~ , y ~ ) (\widetilde{\mathbf x},\widetilde{\mathbf y}) ( x , y ) n n n x ~ \widetilde{\mathbf x} x y ~ \widetilde{\mathbf y} y t 1 ≥ t 2 ≥ ⋯ ≥ t r t_1\ge t_2 \ge \cdots\ge t_r t 1 ≥ t 2 ≥ ⋯ ≥ t r λ ~ 1 ≥ λ ~ 2 ≥ ⋯ ≥ λ ~ p ∧ q \widetilde\lambda_1 \ge\widetilde\lambda_2\ge\cdots\ge\widetilde\lambda_{p\wedge q} λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ∧ q x \mathbf x x y \mathbf y y z \mathbf z z t c ∈ ( 0 , 1 ) t_c \in(0, 1) t c ∈ ( 0 , 1 ) 1 ≤ i ≤ r 1\le i \le r 1 ≤ i ≤ r t i < t c t_i < t_c t i < t c λ ~ i \widetilde\lambda_i λ i λ + \lambda_+ λ + t i > t c t_i>t_c t i > t c λ ~ i \widetilde\lambda_i λ i t i t_i t i θ i ∈ ( λ + , 1 ) \theta_i \in (\lambda_+,1) θ i ∈ ( λ + , 1 ) x \mathbf x x y \mathbf y y t i t_i t i t c t_c t c t i t_i t i
引用
@article{arxiv.2102.03297,
title = {Sample canonical correlation coefficients of high-dimensional random vectors with finite rank correlations},
author = {Zongming Ma and Fan Yang},
journal= {arXiv preprint arXiv:2102.03297},
year = {2022}
}
备注
Bernoulli Journal (to appear)