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We establish statistical properties of random-weighting methods in LASSO regression under different regularization parameters $λ_n$ and suitable regularity conditions. The random-weighting methods in view concern repeated optimization of a randomized objective function, motivated by the need for computational approximations to Bayesian posterior sampling. In the context of LASSO regression, we repeatedly assign analyst-drawn random weights to terms in the objective function (including the penalty terms), and optimize to obtain a sample of random-weighting estimators. We show that existing approaches have conditional model selection consistency and conditional asymptotic normality at different growth rates of $λ_n$ as $n \to \infty$. We propose an extension to the available random-weighting methods and establish that the resulting samples attain conditional sparse normality and conditional consistency in a growing-dimension setting. We find that random-weighting has both approximate-Bayesian and sampling-theory interpretations. Finally, we illustrate the proposed methodology via extensive simulation studies and a benchmark data example.