Limiting Behavior of Maxima under Dependence

Klaus Herrmann, Marius Hofert, Johanna G. Neslehova
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Abstract

Weak convergence of maxima of dependent sequences of identically distributed continuous random variables is studied under normalizing sequences arising as subsequences of the normalizing sequences from an associated iid sequence. This general framework allows one to derive several generalizations of the well-known Fisher-Tippett-Gnedenko theorem under conditions on the univariate marginal distribution and the dependence structure of the sequence. The limiting distributions are shown to be compositions of a generalized extreme value distribution and a distortion function which reflects the limiting behavior of the diagonal of the underlying copula. Uniform convergence rates for the weak convergence to the limiting distribution are also derived. Examples covering well-known dependence structures are provided. Several existing results, e.g. for exchangeable sequences or stationary time series, are embedded in the proposed framework.
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依赖性下最大值的极限行为
研究了同分布连续随机变量依存序列最大值的弱收敛性,它是在一个相关的 iid 序列的归一化序列的子序列下产生的。在这个一般框架下,我们可以根据序列的单边边际分布和依存结构条件,推导出著名的费希尔-蒂佩特-格内登科定理的几个一般化。极限分布被证明是广义极值分布和扭曲函数的组合,扭曲函数反映了基础协方差对角线的极限行为。此外,还推导出了弱收敛到极限分布的均匀收敛率。一些现有的结果,如可交换序列或静态时间序列的结果,都被嵌入到了所提出的框架中。
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