Strong Law For Arrays Of Random Variables

Probability limit theory is one of the main branchs of probability theory and important basis of mathematic statistics. The central limit theorem was the central subject of probability theory study at 1820s. The classical limit theory is an important achievement in the history of the development of probability theory. The study of modern limit theory is still popular so far, it not only deepened a lot of the basic results of classical theory, but also greatly expanded their field of the study. These are associated with the other branchs of probability theory and the latest developments of mathematical statistics.This paper discusses strong law for arrays of random variables, some new results are obtained. In the first chapter, I introduce some basis concepts and important lemmas. In the second chapter, I discuss an analogue of Kolmogorov's law of the iterated logarithm for arrays, the sufficient conditions of the law of the single logarithm for array of independent random variables are obtained, which extend some well-known results of the Sung[9] and Sung[11]. In the third chapter, the sufficient and necessary conditions of the law of the single logarithm for linear processes generated by array of independent identically distributed random variables are obtained, which extend some well-known results. In the fourth chapter, on the basis of the third chapter, we obtain strong approximations, the law of the single logarithm and the functional law of the single logarithm for linear processes generated by arrays of i.i.d random variables, and show that they are equivalent to the same moment conditions.