Some limit theorems on uncertain random sequences

An uncertain random variable is a measurable function on the chance space. It is used to describe the mixing phenomena with both randomness and uncertainty. The uncertain random sequence is a sequence of uncertain random variables indexed by integers. Three types of convergence concept of uncertain random sequence have been defined, namely, convergence in distribution, convergence almost surely and convergence in measure, and some convergence theorems have been obtained. The main purpose of this paper is to provide some limit theorems on uncertain random sequences. First, we construct two examples to illustrate the concepts of convergence almost surely and convergence in measure for a sequence of uncertain random variables. Then an inequality for uncertain random variable is presented, which states the relationship among chance measure, probability and uncertain measure. Several theorems about convergence of uncertain random sequences are obtained by Borel-Cantelli lemma which is given based on the properties of limit superior. Finally, a convergence theorem for uncertain random series is established. The main results of this paper contain the relevant conclusions for random sequence and uncertain sequence.