The goal of this paper is to investigate the tools of extreme value theory originally introduced for discrete time stationary stochastic processes (time series), namely the tail process and the tail measure, in the framework of continuous time stochastic processes with paths in the space \(\mathcal {D}\) of càdlàg functions indexed by \(\mathbb {R}\), endowed with Skorohod’s J₁ topology. We prove that the essential properties of these objects are preserved, with some minor (though interesting) differences arising. We first obtain structural results which provide representation for homogeneous shift-invariant measures on \(\mathcal {D}\) and then study regular variation of random elements in \(\mathcal {D}\). We give practical conditions and study several examples, recovering and extending known results.

本文的目的是在空间中具有路径的连续时间随机过程的框架内，研究最初为离散时间平稳随机过程（时间序列）引入的极值理论工具，即尾过程和尾测度。 (\mathcal {D}\)由\(\mathbb {R}\)索引的 càdlàg 函数，具有 Skorohod 的J ₁拓扑。我们证明这些对象的基本属性被保留，但出​​现了一些细微（虽然有趣）的差异。我们首先获得结构结果，这些结果为\(\mathcal {D}\)上的齐次平移不变测度提供表示，然后研究\(\mathcal {D}\)中随机元素的规则变化. 我们给出了实际条件并研究了几个例子，恢复和扩展了已知的结果。