Abstract: In this study, we establish the Marcinkiewicz-Zygmund strong law of large numbers for randomly weighted sums of negatively orthant dependent random variables. The law of the single law of logarithm for randomly weighted sums of negatively orthant dependent random variables is also established. Finally, the results are applied to bootstrap sample means and least-squares estimators in a simple linear regression with random design.

Key words and phrases: Bootstrap sample mean, law of the single logarithm, Marcinkiewicz-Zygmund strong law, randomly weighted sum, regression with random design.