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 case where covariance matrix of regression variables has a KMS structure, settings number predictors, p, is proportional to observations, n. The main result paper derivation exact distribution for suitably centered and normalized squared norm product between predictor matrix, X, outcome variable, Y, i.e., statistic ∥X′Y∥22, under rather unrestrictive assumptions model parameters βj. employ variance-gamma order derive results, which, along with allows us easily define statistic. Additionally, consider specific approximate sparsity parameter vector β perform Monte Carlo simulation study. results suggest that approaches limiting fairly quickly even high variable multi-correlation relatively small suggesting possible applications construction statistical testing procedures real-world data related problems.