Motivated by Ellsberg-type models and problems in mathematical finance,we investigate limit behaviours of two different models: one is the very simple Bernoulli trials with ambiguity(or called Ellsberg-type model),and the other is sub-linear expectations arising from mathematical finance.With a new notion of-convolution for random variables,we show that empirical averages obtained from a large number of trials in both models have the same limit distribution.We also investigate the relation between this limit theorem and the weak law of large numbers for nonadditive probability,and show that they are equivalent under the assumption of-convolution on random variables.Our results generalize well-known laws of large numbers(LLNs),using the proofs that are completely different from those in the existing literature.Finally,we discuss four models which satisfy the assumptions of our main results.