Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.

翻译：受函数型数据日益丰富的启发，我们为$L^2[0,1]$空间中正则变化随机元素$X$的极值建立了通用概率与统计框架。研究置于超阈值框架中，将函数极值定义为$L^2$范数$\|X\|$相对较大的观测值$X$。我们的目标是提出能够将此类极值观测降维至有限维投影的框架。本文贡献有两方面：首先，深入研究了一般可分希尔伯特空间随机量的正则变化概念，并提出了仅涉及实值随机变量随机收敛性的新颖具体刻画；其次，提出了能够反映函数极值主要"方向"的函数型主成分分析概念。我们通过限定有限样本下估计误差的希尔伯特-施密特范数上界，研究了极值函数角分量的经验协方差算子的统计性质。模拟与真实数据上的数值实验验证了本研究。