In this paper, we study the complete convergence and the complete integration convergence for weighted sums of $m$-extended negatively dependent ( $m$-END) random variables under sub-linear expectations space with the condition of $\hat{\mathbb{E}}|X{|}^p⩽C_{\mathbb{V}}(|X{|}^p)<\mathrm{\infty }$, $p>1/\alpha$ and $\alpha >3/2$. We obtain the results that can be regarded as the extensions of complete convergence and complete moment convergence under classical probability space. In addition, the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of $m$-END random variables under the sub-linear expectations space is proved.

In the framework of sublinear expectations, we prove the Marcinkiewicz-Zygmund type weak law of large numbers for an array of row-wise weakly negatively dependent (WND) random variables. Moreover, we obtain the strong law of large numbers for linear processes generated by WND random variables. Our theorems extend the existed achievements of the law of large numbers under sublinear expectations.

In this article, using the Fuk-Nagaev type inequality, we studied general strong law of large numbers for weighted sums of $m$-widely acceptable ( $m$-WA, for short) random variables under sublinear expectation space with the integral condition

$$\hat{\mathbb{E}}\left(f^{-}\left(\left|X\right|\right)\right)\le {\mathrm{C}}_{\mathbb{V}}\left(f^{-}\left(\left|X\right|\right)\right)<\mathrm{\infty }$$

and $Choquet$ integrals existence, respectively, where

$$f\left(x\right)=x^{1/\beta }L\left(x\right)$$

for $\beta >1$, $L\left(x\right)>0$ $\left(x>0\right)$ was a monotonic nondecreasing slowly varying function, and $f^{-}\left(x\right)$ was the inverse function of $f\left(x\right)$. One of the results included the Kolmogorov-type strong law of large numbers and the partial Marcinkiewicz-type strong law of large numbers for $m$-WA random variables under sublinear expectation space. Besides, we obtained almost surely convergence for weighted sums of $m$-WA random variables under sublinear expectation space. These results improved the corresponding results of Ma and Wu under sublinear expectation space.