Precise Asymptotics Of Random Variables

Probability Limit Theory is one of the important branches and also an essential theoretical foundations of science of Probability and Statistics. "The epistemological value of the theorey of probability is revealed only by limit theorems. Moreover, without limit theorems it is impossible to understand the real content of the primary concept of all our sciences-the concept of probability ." which is stated in the classical "Limit distribution for sums of independent random variables" by B. V. Gendenko and A. N. Kolmogrov. Classical limit theory is the signify achievement in the progress of Probability. Strong convergence has become the most important and popular orientations of the current study of Probability Limit Theory. Some significant results have been reached through deep research in this dissertation.Recently, more and more people take an interest in researching of Precise Asymptot-ics. Based on the known results, this dissertation firstly study the Precise Asymptitics of order statistics based on a random sample. Let {X,X₁,X₂, ? ? ? } be an iid random variables, and F be the distribution function of X, which is in the maximum domain of attraction of Frechet distribution. It is a significative attempt to take the research on precise asymptotics in maximum domain of attraction, which have a background of hazard of finance. Then, under some suitable conditions, we obtain the precise asymptotics of them for some special boundary functions and weight functions. At the same time, we can obtain the results of order statistics based on non-random sample by select a special parameter. In order to fine more boundary functions and weight functions and reveal the relations among the boundary function, weight function and convergence rate with a uniform form, under some ordinary conditions which may seem complex, we obtained a commonly result. Obviously, the results not only make the known results as the special case of our results, but also reveal the relations among the boundary function, weight function and convergence rate. Therefore, we can present a great deal of boundary function and weight function.Subsequently, we touched the Generalized Order Statistics. Generalized Order Statistics are a subclass of sequential order statistics, and contains a variety of models of ordered random variables often used in probability and statistics, e.g., ordinary order statistics, record values, k—record values, Pfeifer's record model, progressive type II...