| In1966, Lehmann introduced a special sequence, negative quadrant dependent(NQD) sequence, which covers a wide range. In1984, Newman introduced a newsequence, named linear negative quadrant dependent (LNQD) sequence, it is slightlystronger than NQD sequence. As time goes on, kinds of nature of LNQD sequence areconstantly recognized, and it applies more and more widely. The aim of this paper isto study the convergence properties of LNQD sequence in different fields.In the first part, we discussed the law of large number. Its property has arousedwide interest because of numerous applications. In the past decades, a lot of effort wasdedicated, and obtained many classical results of independent and dependentsequence. A Kolmogorov-type strong law of large numbers of ND random variableswas established by Matula, which is the same as i.i.d. sequence, andMarcinkiewicz-type strong law of large Numbers was obtained by Su and Wang forNA random variable sequence with assumptions of identical distribution; Yang et al.gave the strong law of large Numbers of a general method.In the second part, we studied the central limit theorem. The center limit theoremis a hot topic in limit theory, and it caused many scholars’ attention for its practicalapplication in stochastic simulation. Some important applications for LNQD sequencehave been found. See, for example, Newman who established the central limittheorem for a strictly stationary LNQD process. Wang and Zhang provided uniformrates of convergence in the central limit theorem for LNQD sequence. Ko et al.obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. studied thestrong convergence for weighted sums of LNQD arrays, and so forth.In the last part, we introduced the convergence properties of LNQD sequenceunder different conditions. In this field, many scholars did the related research, forexample, Chandra obtainedL1-convergence for pairwise independent randomvariables under Cesáro uniform integrability (CUI) and strong Cesáro uniformintegrability (SCUI). Landers and Rogge obtained a slight improvement over theresults of Chandra and Chandra and Goswami for the case of nonnegative random variables. Chandra and Goswami introduced a new set of conditions called Cesárointegrability (CI(α)) and strong Cesáro integrability (SCI(α)) for asequence of random variables, which are strictly weaker than CUI and SCUI,respectively, and improved the results of Landers and Rogge and the earlier results.The main structure of master degree theses are as follows:In chapter1, we introduce the background of the research and give the structureof this paper.In chapter2, we introduce the concept of LNQD sequences firstly, and thenobtain the LNQD sequence of strong convergence properties using the probabilityinequality given by Fuk and Nagaev, which generalizes the results in Hu and Taylor(1997).In chapter3, we discuss the center limit theorem of the linear process generalizedby LNQD strictly stationary sequence, which is studied on the basis of the results inNewman (1984).In chapter4, we derive someL_p-convergence and complete convergence of themaximum of partial sum for LNQD random variables when such random variables aresubject to RCI(α) and SRCI(α), which are improvements over the results ofChandra and Goswami (2003) and the earlier results. |
