KARHUNEN-LOèVE Expansions With Its Large Deviation And Small Deviation Of Two Types Of Gauss Processes

In probability and statistics, the Gaussian process is an important stochasticprocess, and research of large deviation and small deviation are the most important partin the field of statistics. Karhunen-Loève expansion of Gaussian process has closelyconnected with many fields. By the research of Karhunen-Loève expansion of Gaussianprocess, we can study the nature of process better, and this will enable us to solve theproblems of the subject areas, such as random signal processing, wavelet transform andsome other subjects related to random process easier.The topic of "Karhunen-Loève expansions with its large deviation and smalldeviation of two types of Gaussian processes", mainly for zero mean Gaussianprocesses, by using Mercer theory and Karhunen-Loève expansion theory, we can yieldKarhunen-Loève expansions of two sided Brownian bridge and a detrended Brownianmotion, and research of the large deviation and small deviation of a detrended Brownianmotion. This paper mainly studies the following problems of three aspects:(1)Karhunen-Loève expansion of two sided Brownian bridge: By the research ofpositive definite of covariance function of two sided Brownian bridge, we can yield thatits KL expansion form is the same as standard Brownian motion.(2)Karhunen-Loève expansion of a detrended Brownian motion: First, weintroduce the covariance function of detrended Brownian motion. Second, by the use ofMercer theorem and KL expansion theorem, we introduce that its KL expansion form isthe same as a generalized Brownian bridge.(3)By using the Karhunen-Loève expansion of a detrended Brownian motion, weestablished the Laplace transform and large deviation and small deviation of this kind ofGaussian process.The structure of this paper as follows: In the introduction of the first chapter, weintroduce the background of Gaussian process and the research status domestic andabroad and some relevant circumstances of this topic; In the second chapter, theresearch of the Karhunen-Loève expansion of two sided Brownian bridge is studied;The Karhunen-Loève expansion of a detrended Brownian motion is yielded in the thirdchapter; In the last chapter, the results of the Laplace transform and large deviation andsmall deviation of a detrended Brownian motion are given.