In this paper, with the notion of independence for random variables under upper expectations, we derive a strong law of large numbers for non-additive probabilities. This result can be seen an extension version of Theorem 3.1 that Chen et al. [A strong law of large numbers for non-additive probabilities. Int J Approx Reason. 2013;54:365–377] yielded. Furthermore, two applications of our result are given.

In this paper, our aim is to obtain the modulus of continuity theorem for G-Brownian motion. It turns out that our theorem is a natural extension of the classical result obtained by Lévy ( 1937 Lévy, P. (1937). Théorie de laddition des variables aléatories indépendantes. Paris: Gauthier-Villars. [Google Scholar] ).

In this paper, under some weaker conditions, we give three laws of large numbers (LLNs) under sublinear expectations (capacities), which extend the LLN under sublinear expectations in Peng ( 2008b Peng, S. (2008b). A new central limit theorem under sublinear expectations. arXiv:0803.2656v1 [math.PR], 18 Mar 2008. [Google Scholar] ) and the strong LLN for capacities in Chen ( 2010 Chen, Z. (2010). Strong laws of large numbers for capacities. arXiv:1006.0749v1 [math.PR], 3 Jun 2010. [Google Scholar] ). It turns out that these theorems are natural extensions of the classical strong (weak) LLNs to the case where probability measures are no longer additive.

In this Note, our aim is to obtain the central limit theorem for capacities induced by sublinear expectations.

In this Note, our aim is to obtain Cramér's upper bound for capacities induced by sublinear expectations.