Central Limit Theorem for Linear Eigenvalue Statistics for a Tensor Product Version of Sample Covariance Matrices

For $$k,m,n\in {\mathbb {N}}$$ k , m , n ∈ N , we consider $$n^k\times n^k$$ n k × n k random matrices of the form $$\begin{aligned} {\mathcal {M}}_{n,m,k}({\mathbf {y}})=\sum _{\alpha =1}^m\tau _\alpha {Y_\alpha }Y_\alpha ^T,\quad {Y}_\alpha ={\mathbf {y}}_\alpha ^{(1)}\otimes \cdots \otimes {\mathbf {y}}_\alpha ^{(k)}, \end{aligned}$$ M n , m , k ( y ) = ∑ α = 1 m τ α Y α Y α T , Y α = y α ( 1 ) ⊗ ⋯ ⊗ y α ( k ) , where $$\tau _{\alpha }$$ τ α , $$\alpha \in [m]$$ α ∈ [ m ] , are real numbers and $${\mathbf {y}}_\alpha ^{(j)}$$ y α ( j ) , $$\alpha \in [m]$$ α ∈ [ m ] , $$j\in [k]$$ j ∈ [ k ] , are i.i.d. copies of a normalized isotropic random vector $${\mathbf {y}}\in {\mathbb {R}}^n$$ y ∈ R n . For every fixed $$k\ge 1$$ k ≥ 1 , if the Normalized Counting Measures of $$\{\tau _{\alpha }\}_{\alpha }$$ { τ α } α converge weakly as $$m,n\rightarrow \infty $$ m , n → ∞ , $$m/n^k\rightarrow c\in [0,\infty )$$ m / n k → c ∈ [ 0 , ∞ ) and $${\mathbf {y}}$$ y is a good vector in the sense of Definition 1.1, then the Normalized Counting Measures of eigenvalues of $${\mathcal {M}}_{n,m,k}({\mathbf {y}})$$ M n , m , k ( y ) converge weakly in probability to a nonrandom limit found in Marchenko and Pastur (Math USSR Sb 1:457–483, 1967). For $$k=2$$ k = 2 , we define a subclass of good vectors $${\mathbf {y}}$$ y for which the centered linear eigenvalue statistics $$n^{-1/2}{{\mathrm{Tr}}}\varphi ({\mathcal {M}}_{n,m,2}({\mathbf {y}}))^\circ $$ n - 1 / 2 Tr φ ( M n , m , 2 ( y ) ) ∘ converge in distribution to a Gaussian random variable, i.e., the Central Limit Theorem is valid.