Convex hulls of regularly varying processes

Yu. Davydov   ; C. Dombry

Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 18, Tome 408 (2012), pp. 154-174

Résumé

We consider the asymptotic behaviour of the compact convex subset $\widetilde W_n$ of $\mathbb R^d$ defined as the closed convex hull of the ranges of independent and identically distributed (i.i.d.) random processes $(X_i)_{1\leq i\leq n}$. Under a condition of regular variations on the law of $X_i$'s, we prove the weak convergence of the rescaled convex hulls $\widetilde W_n$ as $n\to\infty$ and analyse the structure and properties of the limit shape. We illustrate our results on several examples of regularly varying processes and show that, in contrast with Gaussian setting, in many cases the limit shape is a random polytope of $\mathbb R^d$.

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