Extremal shot noises, heavy tails and max-stable random fields.

Clément Dombry (1)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]

We consider the extremal shot noise defined by $$M(y)=\sup\{mh(y-x);(x,m)\in\Phi\},$$ where $\Phi$ is a Poisson point process on $\bbR^d\times (0,+\infty)$ with intensity $\lambda dxG(dm)$ and $h:\bbR^d\to [0,+\infty]$ is a measurable function. Extremal shot noises naturally appear in extreme value theory as a model for spatial extremes and serve as basic models for annual maxima of rainfall or for coverage field in telecommunications. In this work, we examine their properties such as boundedness, regularity and ergodicity. Connections with max-stable random fields are established: we prove a limit theorem when the distribution $G$ is heavy-tailed and the intensity of points $\lambda$ goes to infinity. We use a point process approach strongly connected to the Peak Over Threshold method used in extreme value theory. Properties of the limit max-stable random fields are also investigated.

Mots clés: Extremal shot noises; extreme value theory; max-stable random fields; weak convergence; point processes; heavy tails

Domaines: Probabilités [math.PR]

Clément Dombry. Extremal shot noises, heavy tails and max-stable random fields.. 2009. ⟨hal-00411720v2⟩