Limiting spectral distribution of a new random matrix model with dependence across rows and columns

Abstract

We introduce a random matrix model where the entries are dependent across both rows and columns. More precisely, we investigate matrices of the form $\X=(X_{(i-1)n+t})_{it}\in\R^{p\times n}$ derived from a linear process $X_t=\sum_j c_j Z_{t-j}$, where the $\{Z_t\}$ are independent random variables with bounded fourth moments. We show that, when both $p$ and $n$ tend to infinity such that the ratio $p/n$ converges to a finite positive limit $y$, the empirical spectral distribution of $p^{-1}\X\X^{\T}$ converges almost surely to a deterministic measure. This limiting measure, which depends on $y$ and the spectral density of the linear process $X_t$, is characterized by an integral equation for its Stieltjes transform. The matrix $p^{-1}\X\X^{\T}$ can be interpreted as an approximation to the sample covariance matrix of a high-dimensional process whose components are independent copies of $X_t$.