The Inequalities And Strong Law Of Large Numbers For Demimartingale And Dependent Sequence On Finite Second Moment

The limit theorem is the central subject of the probability limit theory. We use probability inequality in order to obtain a lot of perfect results.In order to study random variable convergence inequality, a lot of inequalities has already been given by many scholars. Hajek and Renyi (1955) gave a important inequality named Hajek-Renyi inequality. Many scholars have improved them and obtained a class of Hajek-Renyi-type inequality and their application. Recently, in 2000, Prakasa Rao (2000) proved the associated random variables convergence by using the Hajek-Renyi-type inequality. Sung (2008) further generalized Hajek-Renyi-type inequality, gave a more extensive application which was better than the results of Prakasa Rao (2000). But there were several mistakes in the proof of main theorems.Chow (1960) have proved the maximal inequality for submartingale which contains Hajek-Renyi inequality. Christofides (2000) have given that maximal inequality for demimartingale and demisubmartingale, Wang (2004) have proved the Doob inequality for PA sequence and demimartingale. Recently, Hu et al. (2009) have obtained the Hajek-Renyi-type inequality and strong growth rates for demimartingale and PA sequence which were better than the results of Sung (2008). In addition, they pointed out some mistakes in the paper of Sung (2008).The strong law of large numbers are used to give under the moment condition (r>1). In this paper, firstly, the inequalities and growth rates for demimartingale and PA sequences on finite frist moment (0