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

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  • Vysotsky, Vladislav

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).

Suggested Citation

  • Vysotsky, Vladislav, 2025. "The isoperimetric problem for convex hulls and large deviations rate functionals of random walks," Stochastic Processes and their Applications, Elsevier, vol. 180(C).
    • Handle: RePEc:eee:spapps:v:180:y:2025:i:c:s0304414924002278
    DOI: 10.1016/j.spa.2024.104519
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    References listed on IDEAS

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    1. Wade, Andrew R. & Xu, Chang, 2015. "Convex hulls of random walks and their scaling limits," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4300-4320.
    2. J. Kuelbs & M. Ledoux, 1998. "On Convex Limit Sets and Brownian Motion," Journal of Theoretical Probability, Springer, vol. 11(2), pages 461-492, April.
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