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In this paper we study the asymptotic normality in high-dimensional linearregression. We focus on the case where the covariance matrix of the regressionvariables has a KMS structure, in asymptotic settings where the number ofpredictors, $p$, is proportional to the number of observations, $n$. The mainresult of the paper is the derivation of the exact asymptotic distribution forthe suitably centered and normalized squared norm of the product betweenpredictor matrix, $\mathbb{X}$, and outcome variable, $Y$, i.e. the statistic$\|\mathbb{X}'Y\|_{2}^{2}$. Additionally, we consider a specific case ofapproximate sparsity of the model parameter vector $\beta$ and perform aMonte-Carlo simulation study. The simulation results suggest that the statisticapproaches the limiting distribution fairly quickly even under high variablemulti-correlation and relatively small number of observations, suggestingpossible applications to the construction of statistical testing procedures forthe real-world data and related problems.

Asymptotic normality in linear regression with approximately sparse structure