The Inequalities Of Demimartingales And N-demimartingales And Limit Theorems

The limit theory is one of the main branches of probability theory and it’s also the basis of other branches of probability theory and mathematical statistics. In this paper, we study the maximal and minimal inequalities of demartingale and F-demi-martingale, Azuma type inequality and Chow type inequality of N-demimartingale. By these inequalities, we study the complete convergence of random variables.In chapter 2, we study the inequalities of demimartingale and demisubmartingale. First, we introduce the relation between demimartingale、demisubmartingale and mar-tingale、submartingale in detail. A martingale {Sn,Fn;n≥1} with the natural choice of σ algebras {Fn,n≥1} is a demimartingale and a submartingale {Sn,Fn;n≥1} with the natural choice of σ algebras {Fn,n≥1} is a demisubmartingale. We also point out that the partial sum of a sequence of mean zero associated random variables is a demimartingale. And the inverse is not true through concrete examples. Secondly, we prove a maximal inequality of demisubmartingale and get two important implicat-ions. We also get the complete convergence theorem of demimartingale. Finally, we obtain the minimal inequalities for nonnegative demimartingale and demimartingale.In chapter 3, we study the exponential inequalities and the convergence for nega-tively associated identically distributed nonnegative random variables. We also study the Chow type inequality for N-demimartingale. First, we introduce the relation betw-een N-demimartingle、N-demisupermartingale and martingale、supermartingale. And that a martingale {Sn,Fn;n≥1} with the natural choice of σ algebras {Fn,n≥1} is a N-demimartingale and a supermartingale {Sn,Fn;n≥1} with the natural choice of σ algebraas {Fn,n≥1} is a demisupermartingale. The partial sum of a sequence of mean zero negatively associated random variables is a N-demimartingale. But we point out that the inverse is not true by some examples. Secondly, we study the expon-ential inequality and the convergence for negatively associated identically distributed nonnegative random variables by the Azuma type inequality of N-demimartingale. And then we get the convergence of it. At last, we study the Chow type inequality of N-demimartingale and indicate that three Chow type inequalities are not true.In chapter 4, we study the inequality of F-demimartingale and introduce the di-fference and relation between F-demimartingale and demimartingale. We generaliz-e the maximal and the minimal inequality of demimartingales to F-demimartingale. By these inequalities, we get the complete convergence of F-demimartingale.