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Asymptotic normality in linear regression with approximately sparse structure

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 $β$ 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.