Complete Convergence of Weighted Sums of <i>ρ</i> Mixing Sequences

Abstract

Probability limit theory is not only one of the main branches of probability theory, but also important base of others branches and mathematical statistics. In this paper, we discuss the laws of large number and complete convergence of mixing sequence. The paper can be divided into the following two parts. In the first part, we introduce Cesáro uniform integrability and equivalent conditions for sequences of random variable. Then, under these conditions, we establish the weak law of large number and convergence for weighted sums of sequences of random variable of mixing sequence. As an especial example, we obtain the weak law of large number or convergence for mixing sequence. From the course, we can find that Cesáro uniform integrability is also an effective tool for researching the weak law of large number of mixing sequence .In the second part, we investigate a complete convergence theorem of weighted sums of mixing sequence, and prove that the existent results are especial examples for it.