The Almost Sure Convergence Of NA Sequence And (?)-mixing Sequence

In 1950s, with the development of the classical limit theories about independentrandom variable, many scientists of the probability statistics raised and discussedkinds of the convergence properties about the mixing sequence. At that time, first, forthe requirement of some statistical questions, such as some samples withoutindependence, some functions of the independent samples also being not independent,secondly, for the needs from theory studying and the dependent properties in otherfields, including the theories of stochastic sequence and time sequential analysis etc,the questions which were about limit theory of the dependent random variable wereraised. NA random variable sequence, being a kind of important dependent randomvariable was discovered in the early eighties, and (?)-mixing random variablesequence was studied by Bradley in 1990. So studying the limit theorems of NAsequence and (?)-mixing sequence, especially, comparing the same and the differencebetween NA or (?)-mixing sequence and independent sequence, is of importancegreatly.Here, let {X_n, n≥1} be a NA or (?)-mixing random variable sequence. By using ofthe truncation methods of random variables and three series theorem of NAor (?)-mixing sequence, the properties of NA or (?)-mixing sequence are discussed,thealmost sure convergences of NA sequence and (?)-mixing sequence are obtained, andsome simple applications are given. In the writing, the theorem 1.2.1 and theorem2.2.1 are about generalizing the theorem 3.1 in the book[4] from independentsequence to NA sequence and p-mixing sequence. And, extending the coefficients oftheⅰ) and theⅱ) in the theorem 3.1 from p_n=1 and p_n=2 to p_n∈(0,1] andp_n∈(1,2]. Furthermore, as to p_n∈[2,+∞), there are the theorem 1.2.2 and theorem2.2.2. Therefore, some classical strong laws of large numbers are generalized.