6533b855fe1ef96bd12afe7b
RESEARCH PRODUCT
Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices
Anna LytovaAnna Lytovasubject
Statistics and ProbabilityMathematics(all)Multivariate random variableGeneral Mathematics010102 general mathematicslinear eigenvalue statisticsrandom matrices01 natural sciencesSample mean and sample covariance010104 statistics & probabilityDistribution (mathematics)Tensor productStatisticssample covariance matricescentral Limit Theorem0101 mathematicsStatistics Probability and UncertaintyRandom matrixEigenvalues and eigenvectorsMathematicsReal numberCentral limit theoremdescription
For $$k,m,n\in {\mathbb {N}}$$ , we consider $$n^k\times n^k$$ random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$ where $$\tau _{\alpha }$$ , $$\alpha \in [m]$$ , are real numbers and $${\mathbf {y}}_\alpha ^{(j)}$$ , $$\alpha \in [m]$$ , $$j\in [k]$$ , are i.i.d. copies of a normalized isotropic random vector $${\mathbf {y}}\in {\mathbb {R}}^n$$ . For every fixed $$k\ge 1$$ , if the Normalized Counting Measures of $$\{\tau _{\alpha }\}_{\alpha }$$ converge weakly as $$m,n\rightarrow \infty $$ , $$m/n^k\rightarrow c\in [0,\infty )$$ and $${\mathbf {y}}$$ is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of $${\mathcal {M}}_{n,m,k}({\mathbf {y}})$$ converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For $$k=2$$ , we define a subclass of good vectors $${\mathbf {y}}$$ for which the centered linear eigenvalue statistics $$n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ $$ converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.
year | journal | country | edition | language |
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2017-01-31 | Journal of Theoretical Probability |