Some Limit Theorems For Mixing Sequences Of Random Variables

Limit theory is one of the key branches of Probability theory, and also the important foundation of other branches. The research purpose about recent Limit theory is to weaken the restrictions of independence and to customize them to reality for easier identification and application. But considered their complexity, enormous problems has not been fingered out. In this paper, some problems are studied and some results are obtained as follows based on analyzing their characters.1. Given some probability exponential inequalities and moment inequality of negatively dependent random sequence, and we apply these inequalities to study almost sure convergence. As a result, we extend some almost sure convergence theorems for independent random variables to the case of negatively dependent random variables; We study the complete convergence for negatively dependent random variables. As a result, we extends some complete convergence theorems for independent random variables to the case of negatively dependent random variables without necessarily imposing any extra conditions. In the last section of chapter 2 we apply obtain some inequalities to discusses logarithm theorems for negatively dependent random sequences. As a result, some Laws of Logarithm for ND Sequences are obtained. Some results of literature become into particular case of our results and be improved.2. We study the almost sure convergence forρ|~-mixing sequences of random variables. As a result, we improve the corresponding results of Yang (1998), Gan (2004), and Wu (2001). And we extend the classical Khintchine-Kolmogorov convergence theorem, Marcinkiewicz strong law of large numbers, the three series theorem, and the almost sure convergence properties etc. for independent sequences of random variables toρ|~-mixing sequences of random variables without necessarily adding any extra conditions; Then, we study the weak convergence and complete convergence forρ|~-mixing random variable sequences. As a result, the classical weak law of large numbers, and Baum and Katz complete convergence theorem etc. forρ|~-mixing sequences of random variables are investigated. The results extend and greatly improve the corresponding results of literature.