Asymptotic shape of the concave majorant of a Lévy process

Bang, David; González Cázares, Jorge; Mijatović, Aleksandar

Annales Henri Lebesgue, Tome 5 (2022), pp. 779-811.

Nous établissons des théorèmes distributionnels limites pour les statistiques de la forme d’un majorant concave (i.e. les fluctuations de sa longueur, son supremum, son temps d’atteinte et sa valeur en $T$) d’un processus de Lévy sur $[0,T]$ lorsque $T\to \infty$. L’ampleur des fluctuations de la longueur et d’autres statistiques, ainsi que leur comportement asymptotique, varient considérablement en fonction de la queue de la mesure de Lévy. L’outil clé dans les preuves est la représentation récente du majorant concave des processus de Lévy à l’aide d’un processus de bâton brisé.

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 $T$) of a Lévy process on $[0,T]$ as $T\to \infty$. 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.

Classification : 60F05, 60G51
Mots clés : concave majorant, convex minorant, limit theorem, stick-breaking process, Lévy process

[AKMV20] Alsmeyer, Gerold; Kabluchko, Zakhar; Marynych, Alexander; Vysotsky, Vladislav How long is the convex minorant of a one-dimensional random walk?, Electron. J. Probab., Volume 25 (2020), 105 | DOI | MR | Zbl

[APRUB11] Abramson, Josh; Pitman, Jim; Ross, Nathan; Uribe Bravo, Gerónimo Convex minorants of random walks and Lévy processes, Electron. Commun. Probab., Volume 16 (2011), pp. 423-434 | DOI | MR | Zbl

[BDM16] Buraczewski, Dariusz; Damek, Ewa; Mikosch, Thomas Stochastic models with power-law tails. The equation $X=AX+B$, Springer Series in Operations Research and Financial Engineering, Springer, 2016 | DOI | MR | Zbl

[Ber93] Bertoin, Jean Splitting at the infimum and excursions in half-lines for random walks and Lévy processes, Stochastic Processes Appl., Volume 47 (1993) no. 1, pp. 17-35 | DOI | MR | Zbl

[BGCM21a] Bang, David; González Cázares, Jorge I.; Mijatović, Aleksandar A Gaussian approximation theorem for Lévy processes, Stat. Probab. Lett., Volume 178 (2021), 109187 | DOI | MR | Zbl

[BGCM21b] Bang, David; González Cázares, Jorge I.; Mijatović, Aleksandar Presentation on “Asymptotic shape of the concave majorant of a Lévy process”, https://youtu.be/b0AOJm-dE3g, 2021 (YouTube video)

[BGT87] Bingham, Nicholas H.; Goldie, Charles M.; Teugels, Jozef L. Regular variation, Encyclopedia of Mathematics and Its Applications, 27, Cambridge University Press, 1987 | DOI | MR | Zbl

[Bil99] Billingsley, Patrick Convergence of probability measures, Wiley Series in Probability and Statistics, John Wiley & Sons, 1999 (a Wiley-Interscience Publication) | DOI | MR | Zbl

[GCM22] González Cázares, Jorge I.; Mijatović, Aleksandar Convex minorants and the fluctuation theory of Lévy processes, ALEA, Lat. Am. J. Probab. Math. Stat., Volume 19 (2022), pp. 983-999 | DOI | Zbl

[GCMUB22] González Cázares, Jorge I.; Mijatović, Aleksandar; Uribe Bravo, Gerónimo Geometrically convergent simulation of the extrema of Lévy processes, Math. Opr. Res., Volume 47 (2022) no. 2, pp. 1141-1168 | DOI | Zbl

[IL71] Ibragimov, Il’dar A.; Linnik, Yuriĭ V. 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

[Kal02] Kallenberg, Olav Foundations of modern probability, Probability and Its Applications, Springer, 2002 | DOI | MR | Zbl

[Kin93] Kingman, John F. C. Poisson processes, Oxford Studies in Probability, 3, Clarendon Press, 1993 (Oxford Science Publications) | MR | Zbl

[KLM12] Kampf, Jürgen; Last, Günter; Molchanov, Ilya On the convex hull of symmetric stable processes, Proc. Am. Math. Soc., Volume 140 (2012) no. 7, pp. 2527-2535 | DOI | MR | Zbl

[MW16] Molchanov, Ilya; Wespi, Florian Convex hulls of Lévy processes, Electron. Commun. Probab., Volume 21 (2016), 69 | DOI | MR | Zbl

[MW18] McRedmond, James; Wade, Andrew R. The convex hull of a planar random walk: perimeter, diameter, and shape, Electron. J. Probab., Volume 23 (2018), 131 | DOI | MR | Zbl

[Nag00] Nagasawa, Masao Stochastic processes in quantum physics, Monographs in Mathematics, 94, Birkhäuser, 2000 | DOI | MR | Zbl

[Pet75] Petrov, Valentin V. Sums of independent random variables, Ergebnisse der Mathematik und ihrer Grenzgebiete, 82, Springer, 1975 (translated from the Russian by A. A. Brown) | MR | Zbl

[Pet95] Petrov, Valentin V. Limit theorems of probability theory, Oxford Studies in Probability, 4, Clarendon Press, 1995 (Sequences of independent random variables, Oxford Science Publications) | MR | Zbl

[PUB12] Pitman, Jim; Uribe Bravo, Gerónimo The convex minorant of a Lévy process, Ann. Probab., Volume 40 (2012) no. 4, pp. 1636-1674 | DOI | MR | Zbl

[RFZ20] Randon-Furling, Julien; Zaporozhets, Dmitry Convex hulls of several multidimensional Gaussian random walks (2020) (https://arxiv.org/abs/2007.02768)

[Sat13] Sato, Ken-iti 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

[WX15a] Wade, Andrew R.; Xu, Chang Convex hulls of planar random walks with drift, Proc. Am. Math. Soc., Volume 143 (2015) no. 1, pp. 433-445 | DOI | MR | Zbl

[WX15b] Wade, Andrew R.; Xu, Chang Convex hulls of random walks and their scaling limits, Stochastic Processes Appl., Volume 125 (2015) no. 11, pp. 4300-4320 | DOI | MR | Zbl