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The extremal coefficient function (ECF) of a max-stable process $X$ on someindex set $T$ assigns to each finite subset $A\subset T$ the effective numberof independent random variables among the collection $\{X_t\}_{t\in A}$. Weintroduce the class of Tawn-Molchanov processes that is in a 1:1 correspondencewith the class of ECFs, thus also proving a complete characterization of theECF in terms of negative definiteness. The corresponding Tawn-Molchanov processturns out to be exceptional among all max-stable processes sharing the same ECFin that its dependency set is maximal w.r.t. inclusion. This entails sharplower bounds for the finite dimensional distributions of arbitrary max-stableprocesses in terms of its ECF. A spectral representation of the Tawn-Molchanovprocess and stochastic continuity are discussed. We also show how to build newvalid ECFs from given ECFs by means of Bernstein functions.

An exceptional max-stable process fully parameterized by its extremal coefficients