In this paper, an uncertain version of Borel-Cantelli lemma is proved. Besides, some convergence theorems of continuous uncertain measure are proposed. Using these theorems, it is proved that the uncertainty distribution function of the uncertain integral on a real function is a normal one.

In this paper, a useful inequality for central moment of uncertain random variables is proved. Based on this inequality, a convergence theorem for sum of uncertain random variables is derived. A Borel–Cantelli lemma for chance measure is obtained based on the continuity assumption of uncertain measure. Finally, several convergence theorems for uncertain random sequences are established.

This paper examines the numerical solutions of neutral stochastic functional differential equation. study establishes discrete Razumikhin-type theorem to investigate exponential stability in mean square sense Euler–Maruyama solution this In addition, Borel–Cantelli lemma and analysis theory are incorporated discuss almost sure for such equations.

and Applied Analysis 3 2. Preliminaries In this section, wewill present some important lemmaswhich will be used to prove the main results of the paper. The first three lemmas come from Sung [1]. Lemma 11 (cf.[1]). Let {a n , n ≥ 1} be a sequence of positive constants with a n /n ↑. Then the following properties hold. (i) {a n , n ≥ 1} is a strictly increasing sequence with an a n ↑ ∞. (ii) ∑∞ n...

In the study of some dynamical systems limsup set a sequence measurable sets is often interest. The shrinking targets and recurrence are two most commonly studied problems that concern sets. However, zero-one laws for usually treated separately proved differently. this paper, we introduce generalized definition can specialize into recurrence; our approach gives unified proof problems.