最大(最小)值的极限分布问题可看作是一类极值问题.近年来，此类问题得到了来自数学、金融、气象、工程、经济学等不同领域学者的关注，被广泛地应用于金融市场的风险评估，气候预测，地质灾害评估，海况预报，保险精算等行业.因此，对于极值问题的研究不仅具有重要的学术价值而且具有强烈的现实意义.本文以数学分析、概率论、最大熵理论以及随机分析为基础，围绕随机变量及其和、概率测度的最大(最小)值的极限分布展开讨论，主要工作如下： 第一章概述了极值问题的研究背景、意义及国内外的研究现状，简要介绍了本学位论文的研究内容及创新点. 第二章分别考虑了独立同分布随机序列和平稳序列的极小-极大模型，利用分布函数的Taylor展开及几个重要的概率公式，我们得到了这两种序列的极限分布，并且分析了极限分布的收敛性.最后用数值算例验证了理论分析. 第三章考虑了运输问题中的两类特殊路径，证明了这两类路径和最大值的极限分布.在假设每条路径是一条有限状态、离散时间、齐次不可约的遍历Markov链的前提下，结合全概率公式、中心极限定理以及Markov链的分解不仅得到了第一类路径和最大值的极值分布，还建立了第二类路径和的极限分布与其和最大值的极限分布之间的等价性.数值算例验证了我们的结果. 第四章证明了以对数似然比的和为统计量且具有一类特殊动态控制线的控制图在监测有限个相关观测点的分布时是最优的.其最优性表现在当误报率不超过某个给定的预设水平或者所有控制图的误报率不超过某个值时，控制图在过程失控时的平均步长(ARL)是所有控制图中最小的.同时我们证明了对具有负的控制线的ARL₀，当观测值的数量趋于无穷时， ARL₁的值近似为1.随后，利用Markov链最长路径和的极限分布与其和最大值的极限分布之间的等价性给出了当正的控制线充分大时ARL₁的估计.最后，数值仿真验证了最优控制图的监测行为. 第五章利用非线性期望的定义，将非线性期望的求值问题转化为求概率测度的极值问题.结合矩方法和最大熵方法证明了满足最大及最小矩条件的概率密度函数存在的充要条件并求出了该问题的最大熵解，分析了解的收敛性；最后利用数值试验求解了二维最大熵密度函数. 关键词：极小-极大模型；极限分布；Markov链；路径和；对数似然比的和；动态控制线；最优控制图；ARL的估计；非线性期望；最大熵方法；数值算例

 Limiting distribution of maximum (minimum) value can be regarded as a kind of extreme value problem. In recent years, this kind of problem has drawn attention of researchers from different fields, such as mathematics, climate prediction, geological disaster assessment, oceanic forecast, insurance actuarial and so on. Therefore, the study of the extreme value problem not only has great significance but also plays an important role in the realistic society. The limiting distributions of maximum (minimum) value of random variables and sums of random variables, probability measure are considered in this paper in the light of mathematical analysis, probability theory, maximum entropy theory and stochastic analysis. The main work accomplished includes the following: In the first chapter, the research background and significance as well as the current situation of studies both at home and abroad of the extreme value problem are introduced. Moreover, the main contents and invitation of this paper are briefly stated. In the second chapter, we consider the minimum-maximum models for the independent and identically distributed random sequence and stationary sequence, respectively. With Taylor's expansions of the distribution functions and some important probability formulas, the limiting distributions for these two kinds of sequences are obtained and convergence analysis is carried out for the limiting distributions. Numerical experiments are conducted to confirm our theoretical analysis. The third chapter dedicates itself to considering two special kinds of path in transport problems and proving the limiting distributions of maximum sums of these two kinds of path on condition that every path is a discrete time, homogeneous and irreducible ergodic Markov chain with a finite number of states. Combining total probability formula and central limit theorem with the decomposition of Markov chain, we not only derive the extreme value distribution for the first kind but also establish the equivalence between limit distributions of a sum and of a maximum sum for the second kind. Numerical experiments are conducted to confirm our results. In the forth chapter, we show that the control chart with the charting statistics of the sum of log likelihood ratios with a special dynamic control limits for detecting the change in distribution of the finite numbers of dependent observations is optimal, in the sense that the optimality of this kind of control chart shows it has the smallest out-of-control average run length (ARL) among all control charts with a given probability of a false alarm no greater than a preset level, or, among those with a given false alarm rate no less than a given value. Moreover, we prove that the ARL₁ is approximately equal to 1 for any ARL₀ with a negative control limit when the number of observations tends to be infinity. By the equivalence of limit distribution between a longest path sum and a maximum sum of Markov chain, we give the estimation of the ARL₁ with a large enough positive control limit. The numerical simulations illustrate the detection performance of the optimal control chart. The last chapter is devoted to converting the evaluation of nonlinear expectation into seeking some probability density functions satisfied the maximum and minimum moment conditions. Combining the moment method and the maximum entropy method to present a necessary and sufficient condition for the existence of probability density functions. Moreover, we carry out a kind of maximum entropy solutions and analyze the convergence for theses solutions. Numerical experiments are presented to compute the two-dimensional maximum entropy density functions. KEY WORDS: min-max model; limiting distribution; Markov chain; path sum; sum of log likelihood ratios; dynamic control limits; optimal control charts; estimate of ARL; nonlinear expectation; maximum entropy method; numerical experiments