高维随机向量样本典型相关系数的极限分布
概率论
2022-06-28 v3
摘要
考虑两个具有有限秩相关的高维随机向量 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 ~ = x + A z \widetilde{\mathbf x}=\mathbf x+A\mathbf z x = x + A z y ~ = y + B z \widetilde{\mathbf y}=\mathbf y+B\mathbf z y = y + B z 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 A ∈ R p × r A\in\mathbb R^{p\times r} A ∈ R p × r B ∈ R q × r B\in\mathbb R^{q\times r} B ∈ R q × r ( x ~ , y ~ ) (\widetilde{\mathbf x},\widetilde{\mathbf y}) ( x , y ) n n n r r r t 1 ≥ ⋯ ≥ t r t_1\ge\cdots\ge t_r t 1 ≥ ⋯ ≥ t r x ~ \widetilde{\mathbf x} x y ~ \widetilde{\mathbf y} y λ ~ 1 ≥ ⋯ ≥ λ ~ r \widetilde\lambda_1\ge\cdots\ge\widetilde\lambda_r λ 1 ≥ ⋯ ≥ λ r r r r 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 ) t i > t c t_i>t_c t i > t c n ( λ ~ i − θ i ) \sqrt{n}(\widetilde\lambda_i-\theta_i) n ( λ i − θ i ) θ i > λ + \theta_i>\lambda_+ θ i > λ + t i t_i t i n ( λ ~ i − θ i ) \sqrt{n}(\widetilde\lambda_i-\theta_i) n ( λ i − θ i ) x \mathbf x x y \mathbf y y z \mathbf z z
引用
@article{arxiv.2103.08014,
title = {Limiting distribution of the sample canonical correlation coefficients of high-dimensional random vectors},
author = {Fan Yang},
journal= {arXiv preprint arXiv:2103.08014},
year = {2022}
}
备注
Electronic Journal of Probability (to appear)