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In this work, the Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of widely orthant dependent random variables is established under mild conditions including infinite variance. As applications, some new results such as the Cesàro strong law of large numbers, the strong consistency of the least squares estimator in multiple linear regression models, and the strong consistency of the wavelet estimator in nonparametric regression models are presented. Simulation studies are also provided to support the theoretical results.

Marcinkiewicz–Zygmund type strong law of large numbers for weighted sums of random variables with infinite moment and its applications