Some Maximal Inequalities For Multi-Dimensional Indexed Demi(sub)martingales And Their Application

As an important research direction for modern probability limit theory,the research on limit theory of multi-dimensional random variables already has attracted the attention of many scholars,and they have gained many classical conclusions.Multi-dimensional sums of random variable sequences is called multi-dimensional martingale,but multi-dimensional demimartingale is a class random variables se-quences which are more extensive than multi-dimensional martingale.Based on the existed probability inequalities for the demi(sub)martingale sequence {Sn,n?1},we firstly explore the probability inequality of dependent random variable sequence for conditional demi(sub)martingale and two-parameter demi(sub)martingale like {cnSn,n?1} and {cng(Sn),n?1} in this paper,then we give a probability inequalities of multi-dimensional demi(sub)martingale.Mean-while,we give a class of multi-dimensional martingale maximal inequalities,in order to obtain the limit theorem for multi-dimensional(demi)martingale.The main works are as follows:Firstly,we give some maximal inequalities of random sequence for conditional demimartingale like {cnSn,n?1} and {cng(Sn),n?1},some of these results generalize and improve the corresponding results in recent papers.Secondly,we establish a class of probability inequalities of two-parameter demi(sub)martingale and multi-dimensional demi(sub)martingale.Thirdly,we give some strong law of large numbers for multi-dimensional martingale sequence {Sn,n?Z+d}.