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

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∈Rp×n derived from a linear process Xt=∑jcjZt-j, where the {Zt} 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-1XXT converges almost surely to a deterministic measure. This limiting measure, which depends on y and the spectral density of the linear process Xt, is characterized by an integral equation for its Stieltjes transform. The matrix p-1XXT can be interpreted as an approximation to the sample covariance matrix of a high-dimensional process whose components are independent copies of Xt.