On a new concept of stochastic domination and the laws of large numbers

Consider a sequence of positive integers {kn,n≥1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{k_n,n\ge 1\}$$\end{document}, and an array of nonnegative real numbers {an,i,1≤i≤kn,n≥1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_{n,i},1\le i\le k_n,n\ge 1\}$$\end{document} satisfying supn≥1∑i=1knan,i=C0∈(0,∞).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sup _{n\ge 1}\sum _{i=1}^{k_n}a_{n,i}=C_0\in (0,\infty ).$$\end{document} This paper introduces the concept of {an,i}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_{n,i}\}$$\end{document}-stochastic domination. We develop some techniques concerning this concept and apply them to remove an assumption in a strong law of large numbers of Chandra and Ghosal (Acta Math Hung 71(4):327–336, 1996). As a by-product, a considerable extension of a recent result of Boukhari (J Theor Probab, 2021. https://doi.org/10.1007/s10959-021-01120-6) is established and proved by a different method. The results on laws of large numbers are new even when the summands are independent. Relationships between the concept of {an,i}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_{n,i}\}$$\end{document}-stochastic domination and the concept of {an,i}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{a_{n,i}\}$$\end{document}-uniform integrability are presented. Two open problems are also discussed.