On the diminishing process of Bálint Tóth
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- by Péter Kevei and Viktor Vígh PDF
- Trans. Amer. Math. Soc. 368 (2016), 8823-8848 Request permission
Abstract:
Let $K$ and $K_0$ be convex bodies in $\mathbb {R}^d$, such that $K$ contains the origin, and define the process $(K_n, p_n)$, $n \geq 0$, as follows: let $p_{n+1}$ be a uniform random point in $K_n$, and set $K_{n+1} = K_n \cap (p_{n+1} + K)$. Clearly, $(K_n)$ is a nested sequence of convex bodies which converge to a non-empty limit object, again a convex body in $\mathbb {R}^d$. We study this process for $K$ being a regular simplex, a cube, or a regular convex polygon with an odd number of vertices. We also derive some new results in one dimension for non-uniform distributions.References
- Gergely Ambrus, Péter Kevei, and Viktor Vígh, The diminishing segment process, Statist. Probab. Lett. 82 (2012), no. 1, 191–195. MR 2863042, DOI 10.1016/j.spl.2011.09.016
- Octavio Arizmendi and Victor Pérez-Abreu, On the non-classical infinite divisibility of power semicircle distributions, Commun. Stoch. Anal. 4 (2010), no. 2, 161–178. MR 2662723, DOI 10.31390/cosa.4.2.03
- François Bavaud, Adjoint transform, overconvexity and sets of constant width, Trans. Amer. Math. Soc. 333 (1992), no. 1, 315–324. MR 1132431, DOI 10.1090/S0002-9947-1992-1132431-9
- Jean Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, vol. 102, Cambridge University Press, Cambridge, 2006. MR 2253162, DOI 10.1017/CBO9780511617768
- Patrick Billingsley, Probability and measure, 3rd ed., Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1995. A Wiley-Interscience Publication. MR 1324786
- Matthew P. Clay and Nándor J. Simányi, Rényi’s parking problem revisited, Stoch. Dyn. 16 (2016), no. 2, DOI 10.1142/S021949371660006, to appear.
- Charles M. Goldie and Ross A. Maller, Stability of perpetuities, Ann. Probab. 28 (2000), no. 3, 1195–1218. MR 1797309, DOI 10.1214/aop/1019160331
- Ulrich Martin Hirth, Probabilistic number theory, the GEM/Poisson-Dirichlet distribution and the arc-sine law, Combin. Probab. Comput. 6 (1997), no. 1, 57–77. MR 1436720, DOI 10.1017/S0963548396002805
- PawełHitczenko and Gérard Letac, Dirichlet and quasi-Bernoulli laws for perpetuities, J. Appl. Probab. 51 (2014), no. 2, 400–416. MR 3217775, DOI 10.1239/jap/1402578633
- Alfréd Rényi, On a one-dimensional problem concerning random space filling, Magyar Tud. Akad. Mat. Kutató Int. Közl. 3 (1958), no. 1-2, 109–127 (Hungarian, with English and Russian summaries). MR 104284
- Jayaram Sethuraman, A constructive definition of Dirichlet priors, Statist. Sinica 4 (1994), no. 2, 639–650. MR 1309433
- Moshe Shaked and J. George Shanthikumar, Stochastic orders, Springer Series in Statistics, Springer, New York, 2007. MR 2265633, DOI 10.1007/978-0-387-34675-5
- I. M. Yaglom and V. G. Boltyanskiĭ, Vypuklye figury, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1951 (Russian). MR 0049582
Additional Information
- Péter Kevei
- Affiliation: MTA-SZTE Analysis and Stochastics Research Group, Bolyai Institute, University of Szeged, H-6720, Szeged, Aradi vértanúk tere 1, Hungary
- MR Author ID: 834278
- Email: kevei@math.u-szeged.hu
- Viktor Vígh
- Affiliation: Department of Geometry, Bolyai Institute, University of Szeged, H-6720, Szeged, Aradi vértanúk tere 1, Hungary
- Email: vigvik@math.u-szeged.hu
- Received by editor(s): June 25, 2014
- Received by editor(s) in revised form: November 5, 2014, and November 18, 2014
- Published electronically: March 2, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8823-8848
- MSC (2010): Primary 60D05; Secondary 52A22, 60G99
- DOI: https://doi.org/10.1090/tran/6620
- MathSciNet review: 3551590