Strong Limit Theorems For Locally Sub-Gaussian Stochastic Sequence

Probability limit theory is one of the important branches and also is an essential theoretical foundation of probability and statistics. The famous probability scholar Kolmogorov of previous Soviet Union said:"Only probability limit theory can reveal the epistemological value of probability. Without it, you couldn’t understand the real meaning of the fundamental conceptions in probability." Strong convergence has become the most important and popular direction in the current study of probability limit theory. The limit properties for the partial sums of locally Sub-Gaussian stochastic sequence are discussed in this dissertation. By introducing the concept of relative entropy and using limit property of likelihood ratio together with analytic methods, we extend the strong limit theorems expressed by in inequality. Besides, a class of strong limit theorems for locally Sub-Gaussian stochastic sequence represented by inequalities are given, that is strong deviation theorem.In preface, we briefly introduce the background of probability limit theory and development present situation, and introduce the basic ideas and methods of studying strong limit theorems for locally Sub-Gaussian stochastic sequence in this paper.In chapter one, we investigate the strong limit theorems for locally Sub-Gaussian stochastic sequence of two different averages. The class of locally Sub-Gaussian random variables includes that of Sub-Gaussian random variables with v=0and δ=∞.Furthermore, we show that most probability distributions used in practice such as the binomial, normal, Poisson and gamma distributions are locally Sub-Gaussian. Which broadened the scope of Sub-Gaussian random variable.In chapter two, we investigate the strong deviation theorems for locally Sub-Gaussian stochastic sequence. By introducing the concept of relative entropy and using limit property of likelihood ratio together with analytic methods, we give a class of strong limit theorems for locally Sub-Gaussian stochastic sequence represented by inequalities, including some strong deviation theorems of arithmetic average, generalized average and weighted sum average for locally Sub-Gaussian stochastic sequence.In chapter three, we obtain several criterions for locally Sub-Gaussian random variable, and then the integrability of et|F|2of random function Fis discussed, which is defined by locally Sub-Gaussian stochastic sequence.