Consistency Of Kernel Density Estimation For The Random Sequences And Strong Convergence Of Extreme For The Gaussian Process

Probability limit theory primarily studies on various limit behaviors of all kinds ofrandom sequences and stochastic processes. Probability limit theory occupies animportant place in modern probability theory and it is the theoretical basis ofstatistical theories and their applications.An important topic of probability limit theory is the kernel estimation ofprobability density function. In this dissertation, we mainly studied the followingseveral questions about the kernel estimations of probability density functions for twoclasses random variables sequences. First，the kernel estimation of probability densityfunction of the independent identical distribution (iid) random variables sequence isstudied. Here, the iid sequence is made up of the surrogate and validation data. Weprove that the strong convergence of recursive kernel density estimation under somemild conditions. Our achievement expands the results on limit properties of randomvariables sequences based on a validation data sample. Next, the convergence ofKaplan-Meier estimation of probability density function for a random censored modelis considered. In this model, the survival and censoring times form a stationary-mixed random sequence and its cumulative distribution function of the censoringtime is unknown. We obtain a better r-th rate of convergence than that of Wei (2001)under some weaker conditions. After that, the Berry-Esseen bound of the kernelestimation of probability density function is investigated for the-mixed randomlycensored model based on Liang (2009). In this model, the cumulative distributionfunction of the censoring time is known. Although the-mixed sequence is a subsetof the α-mixed sequence，the convergence rate is faster than that of Liang (2009)and some conditions are weaker. Thus, we have got a more favorable Berry-Esseenbound.Another important topic of probability limit theory is the central limit theorem.Through a long period of research and development，there are various forms of central limit theorems and almost sure central limit theorems for the independent ordependent random variables sequences and stochastic processes. Relatively speaking，it is infrequent for the research on the almost sure central limit theorem of stochasticprocesses. Tan (2013) obtained a central limit theorem of the extreme of the stationaryGaussian process in the almost sure sense. Based on this paper，we obtain an wellimproved central limit theorem about the extreme of a stationary Gaussian process inthe almost sure sense. The weight function is expanded and some conditions arereplaced in Tan (2013).