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

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