泛函广义Hill过程及其应用
A functional Generalized Hill process and applications
摘要：
本文研究随机过程序列T_{n}（f）= \sum_{j=1}^{j=k} f（j）（\log X_{n-j+1,n} - \log X_{n-j,n}）的函数渐近性态，它由函数$f:\mathbb{N} \backslash {0} \longmapsto \mathbb{R}_{+}$的某些类$\mathcal{F}$表示，其中$k=k(n)$满足1 \leq k \leq n，k/n \rightarrow 0 \text{as} n \rightarrow \infty .这是一个泛函广义Hill过程，当$F$在极值区域时，它包含尽可能多的极值指标的新估计。在有界实函数空间弱收敛的新背景下，本文重点研究了它的泛函和一致渐近律。接下来将详细说明类$\mathcal{F} $的显式示例的结果。
英文摘要：
We are concerned in this paper with the functional asymptotic behaviour of the sequence of stochastic processes T_{n}(f)=\sum_{j=1}^{j=k}f(j)(\log X_{n-j+1,n}-\log X_{n-j,n}), indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N} \backslash {0} \longmapsto \mathbb{R}_{+}$ and where $k=k(n)$ satisfies 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty. This is a functional generalized Hill process including as many new estimators of the extremal index when $F$ is in the extremal domain. We focus in this paper on its functional and uniform asymptotic law in the new setting of weak convergence in the space of bounded real functions. The results are next particularized for explicit examples of classes $\mathcal{F} $.
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