Chover's Law Of The Iterated Logarithm And The Almost Sure Central Limit Theorem For Associated Sequences

The law of the iterated logarithm is an important achievement of the Probability limit theory and the theorem of strong law of larger numbers. Therefore, the study of the iterated logarithm law is a great interest to scholars of both home and abroad. There have been many classic achievements of independent and dependent sequences, including some study on the law of the iterated logarithm of partial sums and sequences. There is not only close relation between the partial sums and weighted sums, but also essential differences. Recently, the study on the law of the iterated logarithm of weighted sums sequences has become a major topic of the Probability limit theory.Another major topic of the Probability limit theory is the almost sure central limit theorem. Because of its actual application of its random simulation and other aspects, it has attracted the attention of many scholars, and there have been many important achievements of its study.The negative dependent random variables and the strong mixing sequences are two important situations of the independent sequences. The definition of the negative dependent was proposed by Joag-Dev and Proschan in 1983. Because of its wide application in reliability theory and multivariate statistical analysis, people got great interest in it. The strong mixing sequence is a more extensively applicable sequence compared with the dependent random variables which was introduced by Rosenblatt (1956). As its definition tells, the strong mixing sequence is asymptotic independence. In view of its wide application in the random simulation, studies on it have attracted the attention of many scholars. The most famous conclusion to the law of the iterated logarithm is the Law of Iterated Logarithm of Hartman-Wintner by Hartman-Wintner on the condition of the independent identically distributed. On such basis, Kolmogorov removed the limits of identically distributed, broadened the variance value and got the Law of Iterated Logarithm of Kolmogorov of the Kolmogorov type; Chover (1966) got the characteristic index,α∈(0, 2) of independent sequences of the law of the iterated logarithm of the Chover type on the condition of the domain of attraction of a stable distribution; Mikosch (1984) and Vasudeva(1984) proposed other results of the independent sequences of the law of the iterated logarithm of the Chover type. On the basis of the forerunners'studies, Qi and Chen (1996) proposed the independent sequences: the characteristic index,α∈(0, 2) of independent sequences of the law of the iterated logarithm of the Chover type of the on the condition of the domain of attraction of a stable distribution. Wu (2009) canceled the limit of"independent"and extended the achievements of Qi and Chen to the NA independent sequences so as to make the law of the iterated logarithm of the Chover type much more perfect. Chen Yanping (2006) got the law of the iterated logarithm of the Chover type of the random variables of weighted sums and the products of partial sums. The first two chapters of this paper extends the achievements of the independent random variable sequences to the NA situation, and verifies that the negative associated random variables and the independent random variable sequences have the same weighted sums and the products of partial sums of the law of the iterated logarithm of the Chover type.In recent years, more and more scholars made studies on all kinds of the characteristics of sums of partial sums. For example, Qi(2003) proposed the independent nonnegative sequence: the characteristic index,α∈(0, 2) of the almost sure central limit theorem the products of partial sums on the condition of the domain of attraction of a stable distribution. Khurelbaatar,G. and Grzegorz A. R.(2006) proposed the almost sure central limit theorem for the product of partial sums of independent identically distributed sequence. Khurelbaatar,G..(2008) improved the conditions of independent identically distributed and got the almost sure central limit theorem for the product of partial sums of independent identically distributed sequence: the characteristic index,α∈(0, 2) on the condition of the domain of attraction of a stable distribution. Zhang Yong and Yang Xiaoyun (2009) proposed the almost sure central limit theorem for the product of partial sums of NA and LNQD random sequences. Hu Xing and Xu Bin (2007) extended the"independent"to"dependent"and proposed the almost sure central limit theorem for the product of partial sums of theφ? mixing sequences. Jin Jingsen (2007) got the almost sure central limit theorem for the product of partial sums of powerful mixing sequences. Founded on that, the third chapter of this paper extended the results of the product of partial sums of powerful mixing sequences and proposed the almost sure central limit theorem for the product of partial sums of powerful mixing sequences.Listed below is the structure of this paper:Chapter 1 An introduction to some conceptions of NA random variables sequences and the domain of attraction of a stable distribution. On the condition of the domain of attraction of a stable distribution of the characteristic indexα∈(0, 2) , making use of some features of the slowly varying function and approaches of moment inequalities and subsequence, proved the law of the iterated logarithm of the Chover type of the NA random variables sequences weighted sums and got the same conclusion as the situation of"independent".Chapter 2 Studies on the basis of chapter 1, discussing the products of partial sums of the law of the iterated logarithm of the Chover type. Making use of choosing the logarithm of the product and turning it into the sum, the author extended the law of the iterated logarithm of the Chover type of the random variables sequences by Chover and got the the products of partial sums of the law of the iterated logarithm of the Chover type. Chapter 3 An introduction to the conception of powerful mixing sequences. Using the relationship between the mixing coefficientα( n)and covariance, limiting the conditions on the mixing coefficientα( n)and making use of the enlightens of studies of chapter 2, the author extended the results of the almost sure central limit theorem for the product of partial sums of powerful mixing sequences and got the almost sure central limit theorem for the product of partial sums of powerful mixing sequences.