The Inequalities Of Demimartingales And Other Random Sequences And Their Applications

Probability theory is a science of studying the law of random phenomena, which is full of nature and people’s live. With the progress of human society, the increasingly fast development of science, technology and the global econ-omy, probability theory plays a more and more important role in many fields, obtains various applications and receives more and more power of develop-ment. The probability limit theory is an important branch of probability the-ory. By using truncating method of random variables, the inequalities such as Jensen’s inequality, Holder’s inequality, Cr inequality, Minkowski’s inequality, exponential-type inequality, Rosenthal-type inequality, maximal-type inequal-ity and so on, this Ph.D. thesis is to investigate some probability inequalities of dependent random variables, and gives the related applications.First, we will investigate some probability inequalities of Demimartingales and N-demimartingales. It can be found that a martingale with the natural choice of σ-algebras and the partial sums of mean zero associated random vari-ables are demimartingales, but the converse statement cannot always be true. Similarly, it is trivial to verify that a martingale with the natural choice of σ-algebras and the partial sums of mean zero negatively associated random variables form N-demimartingales, but the converse statement cannot always be true. So, a martingale with the natural choice of σ-algebras is not only a Demimartingale but also an N-demimartingale. The theory of martingale has been well studied. Various classical results such as Marshall inequality, which is accurate than Kolmogorov’s inequality, concave function inequalities for martingales, convex functions inequalities for martingales, which can be applied to obtain Doob’s inequality, have been obtained. We generalize some results of martingales to the cases of Demimartingales and N-demimartingales, and obtain some maximal type inequalities such as Marshall type inequali-ties, concave function inequalities and convex functions inequalities for these stochastic process. Meanwhile, we investigate some minimal type inequalities for Demimartingales and firstly get the minimal Marshall type inequalities for demimartingales. Last, we show out some similar and different properties and results of Demimartingales and N-demimartingales. Our results generalize the corresponding results of Aebeko, Garsia, Harremoes, Iksanov and Marynych and Mu and Miao.Second, let{Zn}n>1be nonnegative random variables, whose truncated n random variables satisfy the Rosenthal-type inequality. Denote Xn=Mn-1∑Zi, i=1where{Mn}n>1is a sequence of positive real numbers. Under some suitable conditions, the inverse moment can be asymptotically approximated by for all a>0and a>0. It is said this model as inverse moment model I. Meanwhile, with some other conditions, the growth rate is presented as In addition, with a function f(x) satisfying certain conditions, we show that the inverse moment can be asymptotically approximated by E[f(Xn)]-1[f(EXn)]-y, n�'∞. On the other hand, denote Under some conditions, inverse moment can be asymptotically approximated by and the growth rate is presented as It is said this model as inverse moment model Ⅱ. We point out that the inverse moment models Ⅰ and Ⅱ cannot imply each other, and the two models may be applied in different practical applications. Similarly, with a function f(x) satisfying certain conditions, we obtain that n�'∞.. Some examples are illustrated that our results of inverse moment models Ⅰ and Ⅱ are easy to compute the corresponding inverse moments. Our results of inverse moment models Ⅰand Ⅱ generalize and improve the corresponding ones of Shi et al., Sung, Wang et al. and Wu et al.Third, with some simple conditions and weak mixing coefficients, we in-vestigate the asymptotic properties of sample quantiles under the case of α-mixing sequence. For example, let β>3and α(n)=O(n-β), we study the Bahadur representation of sample quantiles and obtain the grown rate as O(n-1/2(bologn·logn)1/2). In addition, let δ>0,0> max{3+5/(1+δ),1+2/δ} and α(n)=O(n-β), we have the rate as O(n-3/4+δ/4(2+δ)(loglogn·logn)1/2). On the other hand, if the mixing coefficients satisfy α(n)=O(n-7/4), then the Berry-Esseen bound of the sample quantiles is presented as O(n-1/9). Mean-while, by strengthening the mixing coefficients to α(n)=O(n-39/11), we obtain the Berry-Esseen bound such as O(n-1/6·logn). Our results of the Bahadur representation and Berry-Esseen bounds of sample quantiles under the case of a-mixing sequence generalize the corresponding ones of Wang et al. and Lahiri and Sun respectively.Last, we investigate the least squares estimator of unknown parametric θ in nonlinear regression model when the errors{ξn}n≥1satisfy some general conditions. Based on moment information of errors{ξn}n≥1,some probability inequalities of the least squares estimator θn of unknow parametric θ for this model are presented. For example, under some conditions, for all p>0and n≥1, it has Meanwhile, with p>1, some examples are shown that our results can be also used or can be got a smaller bound for the cases of errors{ξn}n≥1satisfy-ing respectively. On the other hand, for the cases of errors{ξn}n≥1such as ρ-mixing sequence, NOD sequence, AANA sequence and Lp-mixingale sequence, we also obtain some inequalities of the estimator of nonlinear regression model for least squares estimator θn of un-known parametric θ. Our results generalize and improve some corresponding ones of Prakasa Rao.