The Convergence In P-distance Of Uncertain Sequences

Uncertainty theory is an important branch of mathematics for modeling belief degree in the real world.In uncertainty theory,we discuss some research results about distance and convergence,and then pointed out that the distance of uncertain variables doesn't meet the triangle inequality of mathematical axioms of distance.In order to overcome this shortage,a new definition of distance of uncertain variables is put forward which is -distance,and the properties of convergence of uncertain variables sequence are researched.The work of this paper enriched the contents of uncertainty theory,the details are as follows:First of all,some properties of expected value for uncertain variables are discussed,which provides theoretical foundation for -distance.Secondly,a new definition of distance of uncertain variables is presented which is -distance.Based on -distance,the concept of convergence in -distance of uncertain variables sequence is put forward.Then,the metric space of uncertain variables is defined,and the completeness of this space is proved.Furthermore,we give the concept of -distance of uncertain vectors and corresponding metric space,and then investigate some properties of them.Finally,some properties of convergence in measure and convergence in -distance are discussed.Then,we present a new kind of sequence convergence which is complete convergence,and discuss the relationships among these convergence concepts.