The tail process and tail measure of continuous time regularly varying stochastic processes

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