  In this paper we study the asymptotic normality in high-dimensional linear
regression. We focus on the case where the covariance matrix of the regression
variables has a KMS structure, in asymptotic settings where the number of
predictors, $p$, is proportional to the number of observations, $n$. The main
result of the paper is the derivation of the exact asymptotic distribution for
the suitably centered and normalized squared norm of the product between
predictor matrix, $\mathbb{X}$, and outcome variable, $Y$, i.e. the statistic
$\|\mathbb{X}'Y\|_{2}^{2}$. Additionally, we consider a specific case of
approximate sparsity of the model parameter vector $\beta$ and perform a
Monte-Carlo simulation study. The simulation results suggest that the statistic
approaches the limiting distribution fairly quickly even under high variable
multi-correlation and relatively small number of observations, suggesting
possible applications to the construction of statistical testing procedures for
the real-world data and related problems.
