The Laws Of Large Numbers For Set-valued Negatively Dependent Random Variables

The classical limit theorems of single-valued random variables have got a lot of beautiful results which are obtained in independent case. But it is not always plausible to assume that the sequence of random variables are independent in many stochastic models. Sometimes the increasing of one random variable will induce the decreasing of another random variable. Then the concept of negatively dependent are useful. In real life, however, because of uncertainty of event’s development, single-valued random variables which can’t perfectly describe event. But set-valued random variables can well describe that. The theory of set-valued random variables is a natural extension of that of general single-valued random variables. Limit theorems play an important role in probability, which include laws of large numbers, central limit theorems,large deviation, etc. This thesis is mainly concerned with two parts. One part is about the weak laws of large numbers for weighted sums of set-valued negatively random variables. Another is about the strong law of large numbers for weighted sums of set-valued negatively random variables.The first part is about the weak laws of large numbers for weighted sums of set-valued random variables. The convergence is in the sense of the Hausdorff metric. Firstly, similar to the definition of negatively dependent of single-valued random variables, we define negatively dependent of set-valued random variables, and discuss the properties. We all know that the interval-valued random variable is a special of set-valued random variable. So, we discuss the properties of interval-valued negatively dependent random variables, list some properties, and prove the weak laws of large numbers for weighted sums of interval-valued random variables in the sense of Hausdorff metric. Finally, we obtain the weak laws of large number for weighted sums of set-valued random variables in the sense of Hausdorff metric. The results are the generalization of those by Bozorgnia, Patterson, Taylor in 1992 and 1993.The second part is about the strong law of large numbers for weighted sums of set-valued random variables.The convergence is in the sense of the Hausdorff metric. We firstly introduce the definition of Hukuhara difference between two sets. We also introduce the Huasdorff metric which is defined by the support function. Meanwhile, we discuss the properties of support function. The last but the most important, we prove the strong law of large numbers for weighted sums of set-valued random variables.