The Probability Inequalities And Limit Theorems For Demimartingale And Three Dependent Sequences

The limit theory of dependent suquences has wide applications in applied probability, statistics, insurance and financial mathematics, complex systems, reliability theory, sur-vival analysis and other fields. This thesis focuses mainly on probability inequalities for demimartingale and three dependent sequences, such as Bernstein-type inequality, Hajek-Renyi-type inequality, Chow-type maximal inequality, Doob-type maximal inequality, and so forth. The limit theorems for demimartingale and three dependent sequences, such as complete convergence, almost sure convergence, strong growth rate, and so on, are also considered.In Chapter 2, we study the moment inequality forφmixing suquence. By the moment inequality, we obtain the Kolmogorov-type convergence theorem, three series theorem, strong law of large numbers and strong growth rate forφmixing suquence. We also give some new results of probability inequalities forφmixing suquence, such as Hajek-Renyi-type inequality, and so on. As a consequence, the integrability of supremum forφmixing suquence can be proved. By the Bernstein-type inequality, we investigate the complete convergence and asymptotic approximation of inverse moment forφmixing suquence. The asymptotic approximation of inverse moment that we obtain generalizes and improves the results of Theorem 3 in Kaluszka and Okolewski, Theorems 2.1 and 2.3 in Hu et al. and Theorem 1 in Wu et al.(the condition sup1≤i≤n EZi/Bn≤C1 can be removed). In addition, we point out that there ia a mistake in the proof of the main results in Sun and Ling, and give the Bahadur representation for sample quantiles underφmixing sequence. We get better bound than that in Ling.In Chapter 3, we investigate the exponential inequalities and strong convergence properties for bounded and unbounded NOD sequence. NOD sequence is a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases. The main purpose of this chapter is to establish the following exponential inequality for unbounded NOD sequence under suitable moment conditions: where is a sequence of positive numbers satisfying Furthermore, we study the strong convergence properties for NOD sequence, which gener-alize and improve the corresponding results for NA sequence in Kim and Kim, Nooghabi and Azarnoosh and Xing et al.The probability inequalities and strong growth rate for LNQD sequence is the main target of Chapter 4. LNQD sequence is also a very broad class of dependent sequences, which contains independent sequence and NA sequence as special cases, but different from NOD sequence. By the Basic properties of LNQD sequence, we get some exponential inequalities, such as Bernstein-type inequality, and so forth, which can be applied to study the complete convergence and almost sure convergence. We also obtain the large deviations and Hajek-Renyi-type inequality from the moment inequality of LNQD suequence, which can be applied to prove the strong growth rate and the integrability of supremum. The results listed above are new results for LNQD suquence.In the lase chapter of this thesis, we study the maximal inequalities and limit the-orem for demimartingale and its convex function. The concept of demimartingale was introduced in 1980s, which contains martingale as a special case. The partial sums of sequences of mean zero independent random variables, mean zero associated random vari-ables and mean zero strongly positive dependent random variables are all demimartingale. We make the following five contributions in this chapter:●We study the Chow-type maximal inequality and Doob-type maximal inequality for demi(sub)martingale and its convex function. Some new results of probability inequalities are also obtained; ●We establish the Doob-type inequality for demi(sub)martingale and its convex function under the case of 01, which generalizes the corresponding results of Christofides and Wang;●We point out that there is a mistake in the proof Theorem 4 in Harremoes, and give a complete proof. A new minimal inequality for demimartinge is also obtained, which can be applied to prove the folowing inequality: where {Sn,n≥1} is a nonnegative demimartingale, S1=1,γ(x)=x-1-logx,x>0. The result above generalizes the corresponding one of Harremoes;●We give some new kinds of strong law of large numbers and strong growth rate for demi(sub)martingale and its convex function, which generalize and improve the corre-sponding ones of Christofides, Chow and Prakasa Rao;●An equivalent condition of uniform integrability for demisubmartingales is pre-sented.