Convergence Properties Of Pairwise NQD Sequences And PA Sequences

In this paper, law of large numbers and complete convergence for pairwise NQD sequences and PA sequences are investigated. It is divided into three chapters as follows: In the first chapter, we give some concepts of the associated sequences including pairwise NQD sequences and PA sequences, summarize the results of their limit theory gained by domestic and foreign scholars, and indicate their theoretical and practical value. In the second chapter, we define the rowwise and pairwise NQD random variables array; introduce Cesáro uniform integrability of random variables array and its equivalent conditions. Then, under certain conditions about Cesáro uniform integrability, we discuss the weak law of large numbers and L~r convergence for the weighted sums of rowwise and pairwise NQD arrays. As corollaries, we obtain some weak laws of large numbers or L~r convergence for pairwise NQD sequences. Our discussions show that Cesáro uniform integrability is also an effective tool in studying the weak law of large numbers for pairwise NQD sequences. Furthermore, we obtain a L~r convergence for pairwise NQD sequence under some suitable conditions. Last, we get a theorem of complete convergence for the weighted sums of pairwise NQD sequence, and an existed result becomes its particular case. In the third chapter, firstly, based on the three series theorem for PA sequences, established by Prakasa, we give general moment conditions satisfied the strong law of large numbers for PA sequences. As corollaries, we obtain some strong laws of large numbers for PA sequences, which are similar to the case of independent sequences. Secondly, by using a moment inequality of maximum partial sum of PA sequences, we study a more general form of complete convergence for the partial sum of PA sequences, and extend the corresponding result of independent sequences.