Types Of Dependent Mixed Random Variables Of The Law Of Large Numbers, And L ~ R Convergence

Limit theory is one of the key branches of probability theory, and also the important foundation of other branches of probability theory and statistics. In the middle 1950s, after the classical limite theory of independent random variables obtained a more comprehensive development, the concept of dependency of random variables has been proposed in some branches of probability theory and statistics. Due to two reasons, one is for the requirement of some statistical questions, such as some sample without independence, or some function of the independent samples also being not independent; The other is for the needs from the theory studying and the requirements of dependency in other fields, such as in Markov chain, theory of random fields and analysis of time series etc.The law of large numbers and convergence are the core theorems of probability limit theory. There are a series of convergence concepts in probability theory, such as convergence in probability, convergence in distribution, almost sure convergence etc. On the basis of former study, we discussed the rowwise NA random variables arrays,the rowwise and pairwise NQD random variables arrays,ρmixing sequences andφ-mixing sequences.The main work of this paper is prescribed as follows. First, under Cesaro uniform integrable conditions, we extended the law of large numbers and convergence of independent case to dependent random variables, these results riched the conclusions of the law of large numbers and convergence. Second, by weakening the relevant conditions, obtained and proved the law of large numbers and convergence of dependent random variables. The basic results were improved and developed.There are four chapters in this paper:In chapter one, we discussed the convergence of rowwise NA random variables arrays. We obtained the Lr convergence and the weak laws of large numbers for weighted sums of rowwise NA random variables arrays under Cesaro uniform integrable conditions and the complete convergence of weighted sums for rowwise NA random variables arrays under conditions weaker than Cesaro uniform integrable conditions.In chapter two, we discussed the convergence of weighted sums for rowwise and pairwise NQD random variables arrays. At present, there are lack of results of Lr convergence for rowwise and pairwise NQD random variables arrays, but most of the results are obtained under 1≤r< 2. By using the Kolmogorov-type inequality to the case of pairwise NQD random sequences, we obtained a L2 convergence of weighted sums for rowwise and pairwise NQD random variables arrays under sup(?)which weaker than Cesaro uniform integrable conditions.The results improved and developed the basic conclutions of the Lr convergnce of weighted sums for random variables arrays.In chapter three,we discussed the strong laws of large numbers ofρmixing sequences.By using the three series theorem ofρmixing sequences,we obtained and proved the strong law of large numbers ofρmixing sequences,wich were similar with the case of independent random variable sequences.In chapter fou, we discussed the weak law of large numbers and convergence ofφ-mixing sequences. First,we obtained the Lr convergence and the weak laws of large numbers ofφ-mixing sequences under Cesaro uniform integrable conditions by using the truncation methods and moment inequalities.Then,As chapter two,we weakened the 2-th Cesaro uniform integrable conditions to (?),and also btained and proved the Lr convergence and the weak law of large numbers ofφ-mixing sequences.Final,we studied the complete convergence ofφ-mixing sequences.These results were the same as those in the independent case.