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Implicit Extremes and Implicit Max-Stable Laws

Abstract

Let X1,...,XnX_1,...,X_n be iid random vectors and f0f\ge 0 be a non-negative function. Let also k(n)=Argmaxi=1,...,nf(Xi)k(n) = {\rm Argmax}_{i=1,...,n} f(X_i). We are interested in the distribution of Xk(n)X_{k(n)} and their limit theorems. In other words, what is the distribution the random vector where a function of its components is extreme? This question is motivated by a kind of inverse problem where one wants to determine the extremal behavior of XX when only explicitly observing f(X)f(X). We shall refer to such types of results as to implicit extremes. It turns out that, as in the usual case of explicit extremes, all limit implicit extreme value laws are implicit max-stable. We characterize the regularly varying implicit max-stable laws in terms of their spectral and stochastic representations. We also establish the asymptotic behavior of implicit order statistics relative to a given homogeneous loss and conclude with several examples drawing connections to prior work involving regular variation on general cones.

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