Some observations on convex hulls of stable random walks

Stjepan Sebek
University of Zagreb, Croatia

Date(s) : 27/09/2022   iCal
14 h 30 min - 15 h 00 min

We consider convex hulls of random walks whose steps belong to the domain of attraction of a stable law in R^d. We prove convergence of the convex hull in the space of all convex and compact subsets of R^d, equipped with the Hausdorff distance, towards the convex hull spanned by a path of the limit stable Lévy process. As an application, we establish convergence of (expected) intrinsic volumes under some mild moment/structure assumptions posed on the random walk. Using the obtained result in the case when the limiting object is a Brownian motion, we develop a closed formula for the expected value of the d-dimensional volume of the convex hull spanned by the time-space trajectory of the (d−1)-dimensional Brownian motion run up to time one.



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