Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation

In this paper, we study the complete convergence and complete moment convergence for weighted sums of extended negatively dependent (END) random variables under sub-linear expectations space with the condition of CV[|X|pl(|X|1/α)]<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C_{\mathbb{V}}[|X|^{p}l(|X|^{1/\alpha})]<\infty$\end{document}, further Eˆ(|X|pl(|X|1/α))≤CV[|X|pl(|X|1/α)]<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\hat{\mathbb {E}}(|X|^{p}l(|X|^{1/\alpha}))\leq C_{\mathbb{V}}[|X|^{p}l(|X|^{1/\alpha })]<\infty$\end{document}, 1<p<2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1< p<2$\end{document} (l(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l(x)>0$\end{document} is a slow varying and monotone nondecreasing function). As an application, the Baum-Katz type result for weighted sums of extended negatively dependent random variables is established under sub-linear expectations space. The results obtained in the article are the extensions of the complete convergence and complete moment convergence under classical linear expectation space.