The isoperimetric problem for convex hulls and large deviations rate functionals of random walks

Vladislav Vysotsky

Stochastic Processes and their Applications, 2025, vol. 180, issue C

Abstract: We study the asymptotic behaviour of the most likely trajectories of a planar random walk that result in large deviations of the area of their convex hull. If the Laplace transform of the increments is finite on R2, such a scaled limit trajectory h solves the inhomogeneous anisotropic isoperimetric problem for the convex hull, where the usual length of h is replaced by the large deviations rate functional ∫01I(h′(t))dt and I is the rate function of the increments. Assuming that the distribution of increments is not supported on a half-plane, we show that the optimal trajectories are convex and satisfy the Euler–Lagrange equation, which we solve explicitly for every I. The shape of these trajectories resembles the optimizers in the isoperimetric inequality for the Minkowski plane, found by Busemann (1947).

Keywords: Convex hull; Convexification; Euler–Lagrange equation; Inhomogeneous isoperimetric problem; Random walk; Rate functional; Rearrangement inequality; Smoothness of minimizers (search for similar items in EconPapers)
Date: 2025
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0304414924002278
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:spapps:v:180:y:2025:i:c:s0304414924002278

Ordering information: This journal article can be ordered from
http://http://www.elsevier.com/wps/find/supportfaq.cws_home/regional
https://shop.elsevie ... _01_ooc_1&version=01

DOI: 10.1016/j.spa.2024.104519

Access Statistics for this article

Stochastic Processes and their Applications is currently edited by T. Mikosch

More articles in Stochastic Processes and their Applications from Elsevier
Bibliographic data for series maintained by Catherine Liu ().