Limit Theorems For Some Mean Dependent Random Variables

Dependent random variable is an important part in probability and statistics with many applications in engineering, economy and medical science and has long been studied in the literature. This research focused on some dependent sequences where the dependence structure involves some factor of the mean value and derived rich limit theorems including basic asymptotics and strong invariance principle. Studying this kind of model is very meaningful since it is associated with many important areas such as clinical trials, adaptive design and recursive stochastic approximation algorithms. The main contents are as follows:In the first part we consider a sequence of dependent Bernoulli variables where the success probability of the trial conditional on the past history is a linear function of the mean number of successes achieved to that point. Under some conditions which are easy to verify, an almost sure invariance principle is established, extending the results of Hedye (2004), James et al. (2008) and Wu et al. (2012). We also generalize the model to a multi-dimensional case and get its central limit theorem. By virtue of a theorem of Zhang (2004), we get a strong approximation result when the parameters do not change with n.The second part is dedicated to a generalized model of Drezner and Farnum (1993) where only finite trials of the past are involved in the dependence structure. This arises from the same motivation as in the famous AR and ARCH models. For the partial sum, we obtain its strong law of large numbers, weak invariance principle and the law of the iterated logarithm. Inspired by Lin et al. (2005), we also derive a strong approximation result with satisfying remainder. Parameter estimation and statistical inference are also studied. Simulation results show that the estimators behave well under small sample size, which is acceptable.In the third part we consider a class of dependent random variables where the dependence structure involves a factor driven by Sn/n. Under very mild conditions for the innovation, we obtain several asymptotic results for the partial sums including basic asymptotics and strong limit theorems. The fact that the asymptotic properties differ strikingly in a neighbour of a critical point makes the model very interesting. As an application, we study a problem involving the unit root test. We also consider the case where the innovation is a linear process and under the sub-linear expectation framework to allow the model for broader applications.