We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at ) of a Lévy process on as . The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the Lévy measure. The key tool in the proofs is the recent representation of the concave majorant for all Lévy processes using a stick-breaking representation.
[BGCM21b] Presentation on “Asymptotic shape of the concave majorant of a Lévy process”, https://youtu.be/b0AOJm-dE3g, 2021 (YouTube video)
[IL71] Independent and stationary sequences of random variables, Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters-Noordhoff Publishing Company, Groningen, 1971, 443 pages (with a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman) | MR | Zbl
[RFZ20] Convex hulls of several multidimensional Gaussian random walks (2020) (https://arxiv.org/abs/2007.02768)
[Sat13] Lévy processes and infinitely divisible distributions, Cambridge Studies in Advanced Mathematics, 68, Cambridge University Press, 2013 (translated from the 1990 Japanese original, revised edition of the 1999 English translation) | MR | Zbl