Since the concept of sub-linear expectation space was put forward, it has well supplemented the deficiency of the theoretical part of probability space. In this paper, we establish the complete convergence and complete integration convergence for weighted sums of widely acceptable (abbreviated as WA) random variables under the sub-linear expectations with the different conditions. We extend the complete moment convergence in probability space to sublinear expectation space.

The aim of this paper is to study and establish precise asymptotics for complete integral convergence in the law of the logarithm under the sub-linear expectation space. The methods and tools in this paper are different from those used to study precise asymptotics theorems in probability space. We extend precise asymptotics for complete integral convergence from the classical probability space to sub-linear expectation space. Our results generalize corresponding results obtained by Fu and Yang^[13]. We further extend the limit theorems in classical probability space.

Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables in a sub-linear expectation space $(\mathrm{\Omega },\mathcal{H},\hat{\mathbb{E}})$. The necessary and sufficient conditions for the convergence rate on the laws of the logarithms and the law of the iterated logarithm are obtained.

In this work, inspired by the extended negatively dependent arrays, we want to obtain a limit theorem on almost sure convergence relying on non-additive probabilities. Meanwhile, we offer two appropriate upper integration conditions as an application, allowing us to derive deterministic bounds based on logarithm. Furthermore, these results extend the limit theorems in classical probability space.

This article aimed to investigate the almost sure convergence theorem of widely negative orthant dependent (WNOD) random variables under sub-linear expectation space. The conclusions in this essay are an extension of the corresponding conclusions in the classical probability space.