Implicit max-stable extremal integrals

Extremal stochastic integrals: a parallel between max–stable processes and α−stable processes

We construct extremal stochastic integrals ∫ e E f(u)Mα(du) of a deterministic function f(u) ≥ 0 with respect to a random α−Fréchet (α > 0) sup–measure. The measure Mα is sup–additive rather than additive and is defined over a general measure space (E, E , μ), where μ is a deterministic control measure. The extremal integral is constructed in a way similar to the usual α−stable integral, but wi...

On Approximating Max-stable Processes and Constructing Extremal Copula Functions

There is an infinite number of parameters in the definition of multivariate maxima of moving maxima (M4) processes, which poses challenges in statistical applications where workable models are preferred. This paper establishes sufficient conditions under which an M4 process with infinite number of parameters may be approximated by an M4 process with finite number of parameters. In statistical i...

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

We consider the extremal shot noise defined by M(y) = sup{mh(y − x); (x,m) ∈ Φ}, where Φ is a Poisson point process on R × (0,+∞) with intensity λdxG(dm) and h : R → [0,+∞] 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 ...

Extremal Stable Graphs

Extremal P4-stable graphs

Call a graph G k-stable (with respect to some graph H) if, deleting any k edges of G, the remaining graph still contains H as a subgraph. For a fixed H, the minimum number of edges in a k-stable graph is denoted by S(k). We prove general bounds on S(k) and compute the exact value of the function S(k) for H = P4. The main result can be applied to extremal k-edge-hamiltonian hypergraphs.