Iterated-logarithm laws for convex hulls of random walks with drift
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- by Wojciech Cygan, Nikola Sandrić, Stjepan Šebek and Andrew Wade;
- Trans. Amer. Math. Soc. 377 (2024), 6695-6724
- DOI: https://doi.org/10.1090/tran/9238
- Published electronically: July 16, 2024
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Abstract:
We establish laws of the iterated logarithm for intrinsic volumes of the convex hull of many-step, multidimensional random walks whose increments have two moments and a non-zero drift. Analogous results in the case of zero drift, where the scaling is different, were obtained by Khoshnevisan [Probab. Theory Related Fields 93 (1992), pp. 377–392]. Our starting point is a version of Strassen’s functional law of the iterated logarithm for random walks with drift. For the special case of the area of a planar random walk with drift, we compute explicitly the constant in the iterated-logarithm law by solving an isoperimetric problem reminiscent of the classical Dido problem. For general intrinsic volumes and dimensions, our proof exploits a novel zero–one law for functionals of convex hulls of walks with drift, of some independent interest. As another application of our approach, we obtain iterated-logarithm laws for intrinsic volumes of the convex hull of the centre of mass (running average) process associated to the random walk.References
- Arseniy Akopyan and Vladislav Vysotsky, Large deviations of convex hulls of planar random walks and Brownian motions, Ann. H. Lebesgue 4 (2021), 1163–1201 (English, with English and French summaries). MR 4353961, DOI 10.5802/ahl.100
- David Bang, Jorge González Cázares, and Aleksandar Mijatović, Asymptotic shape of the concave majorant of a Lévy process, Ann. H. Lebesgue 5 (2022), 779–811 (English, with English and French summaries). MR 4526239, DOI 10.5802/ahl.136
- Gabriele Bianchi, Richard J. Gardner, and Paolo Gronchi, Symmetrization in geometry, Adv. Math. 306 (2017), 51–88. MR 3581298, DOI 10.1016/j.aim.2016.10.003
- Yuan Shih Chow and Henry Teicher, Probability theory, 3rd ed., Springer Texts in Statistics, Springer-Verlag, New York, 1997. Independence, interchangeability, martingales. MR 1476912, DOI 10.1007/978-1-4612-1950-7
- M. Cranston, P. Hsu, and P. March, Smoothness of the convex hull of planar Brownian motion, Ann. Probab. 17 (1989), no. 1, 144–150. MR 972777, DOI 10.1214/aop/1176991500
- Wojciech Cygan, Nikola Sandrić, and Stjepan Šebek, Convex hulls of stable random walks, Electron. J. Probab. 27 (2022), Paper No. 98, 30. MR 4460274, DOI 10.1214/22-ejp826
- Jean-Dominique Deuschel and Daniel W. Stroock, Large deviations, Pure and Applied Mathematics, vol. 137, Academic Press, Inc., Boston, MA, 1989. MR 997938
- Uwe Einmahl, Strong invariance principles for partial sums of independent random vectors, Ann. Probab. 15 (1987), no. 4, 1419–1440. MR 905340
- S. N. Evans, Local Properties of Markov Families and Stochastic Processes Indexed by a Totally Disconnected Field. Ph.D. Thesis, University of Cambridge, 1986.
- Karl Grill, On the average of a random walk, Statist. Probab. Lett. 6 (1988), no. 5, 357–361. MR 933296, DOI 10.1016/0167-7152(88)90013-2
- Peter M. Gruber, Convex and discrete geometry, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 336, Springer, Berlin, 2007. MR 2335496
- Zakhar Kabluchko, Vladislav Vysotsky, and Dmitry Zaporozhets, Convex hulls of random walks: expected number of faces and face probabilities, Adv. Math. 320 (2017), 595–629. MR 3709116, DOI 10.1016/j.aim.2017.09.002
- Jürgen Kampf, Günter Last, and Ilya Molchanov, On the convex hull of symmetric stable processes, Proc. Amer. Math. Soc. 140 (2012), no. 7, 2527–2535. MR 2898714, DOI 10.1090/S0002-9939-2012-11128-1
- Davar Khoshnevisan, Local asymptotic laws for the Brownian convex hull, Probab. Theory Related Fields 93 (1992), no. 3, 377–392. MR 1180706, DOI 10.1007/BF01193057
- J. Kuelbs and M. Ledoux, On convex limit sets and Brownian motion, J. Theoret. Probab. 11 (1998), no. 2, 461–492. MR 1622582, DOI 10.1023/A:1022640007525
- Michel Ledoux and Michel Talagrand, Probability in Banach spaces, Classics in Mathematics, Springer-Verlag, Berlin, 2011. Isoperimetry and processes; Reprint of the 1991 edition. MR 2814399
- Paul Lévy, Le caractère universel de la courbe du mouvement brownien et la loi du logarithme itéré, Rend. Circ. Mat. Palermo (2) 4 (1955), 337–366 (1956) (French). MR 77799, DOI 10.1007/BF02854204
- C. H. Lo, J. McRedmond, and C. Wallace, Functional limit theorems for random walks. Preprint (2018) arXiv:1810.06275.
- Chak Hei Lo and Andrew R. Wade, On the centre of mass of a random walk, Stochastic Process. Appl. 129 (2019), no. 11, 4663–4686. MR 4013876, DOI 10.1016/j.spa.2018.12.007
- Alejandro López Hernández and Andrew R. Wade, Angular asymptotics for random walks, A lifetime of excursions through random walks and Lévy processes—a volume in honour of Ron Doney’s 80th birthday, Progr. Probab., vol. 78, Birkhäuser/Springer, Cham, [2021] ©2021, pp. 315–342. MR 4425798, DOI 10.1007/978-3-030-83309-1_{1}7
- Martin Lotz, Michael B. McCoy, Ivan Nourdin, Giovanni Peccati, and Joel A. Tropp, Concentration of the intrinsic volumes of a convex body, Geometric aspects of functional analysis. Vol. II, Lecture Notes in Math., vol. 2266, Springer, Cham, [2020] ©2020, pp. 139–167. MR 4175761, DOI 10.1007/978-3-030-46762-3_{6}
- M. Lotz and J. A. Tropp, Sharp phase transitions in Euclidean integral geometry. Preprint (2022) arXiv:2208.13919.
- Satya N. Majumdar, Alain Comtet, and Julien Randon-Furling, Random convex hulls and extreme value statistics, J. Stat. Phys. 138 (2010), no. 6, 955–1009. MR 2601420, DOI 10.1007/s10955-009-9905-z
- J. F. W. McRedmond, Convex Hulls of Random Walks. Ph.D. Thesis, Durham University, 2019. http://etheses.dur.ac.uk/13281/.
- James McRedmond and Andrew R. Wade, The convex hull of a planar random walk: perimeter, diameter, and shape, Electron. J. Probab. 23 (2018), Paper No. 131, 24. MR 3896868, DOI 10.1214/18-EJP257
- Z. A. Melzak, The isoperimetric problem of the convex hull of a closed space curve, Proc. Amer. Math. Soc. 11 (1960), 265–274. MR 116263, DOI 10.1090/S0002-9939-1960-0116263-0
- D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities involving functions and their integrals and derivatives, Mathematics and its Applications (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. MR 1190927, DOI 10.1007/978-94-011-3562-7
- P. A. P. Moran, On a problem of S. Ulam, J. London Math. Soc. 21 (1946), 175–179. MR 20799, DOI 10.1112/jlms/s1-21.3.175
- Walter Philipp, Almost sure invariance principles for sums of $B$-valued random variables, Probability in Banach spaces, II (Proc. Second Internat. Conf., Oberwolfach, 1978) Lecture Notes in Math., vol. 709, Springer, Berlin, 1979, pp. 171–193. MR 537701
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- René L. Schilling, Brownian motion—a guide to random processes and stochastic calculus, De Gruyter Textbook, De Gruyter, Berlin, [2021] ©2021. With a chapter on simulation by Björn Böttcher; Third edition [of 2962168]. MR 4368865, DOI 10.1515/9783110741278-202
- Erhard Schmidt, Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet, Math. Ann. 117 (1940), 301–326 (German). MR 3430, DOI 10.1007/BF01450021
- Rolf Schneider, Convex bodies: the Brunn-Minkowski theory, Second expanded edition, Encyclopedia of Mathematics and its Applications, vol. 151, Cambridge University Press, Cambridge, 2014. MR 3155183
- I. J. Schoenberg, An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces, Acta Math. 91 (1954), 143–164. MR 65944, DOI 10.1007/BF02393429
- Timothy Law Snyder and J. Michael Steele, Convex hulls of random walks, Proc. Amer. Math. Soc. 117 (1993), no. 4, 1165–1173. MR 1169048, DOI 10.1090/S0002-9939-1993-1169048-2
- Alexander Yu. Solynin, Exercises on the theme of continuous symmetrization, Comput. Methods Funct. Theory 20 (2020), no. 3-4, 465–509. MR 4175494, DOI 10.1007/s40315-020-00331-y
- Frank Spitzer, Principles of random walk, 2nd ed., Graduate Texts in Mathematics, Vol. 34, Springer-Verlag, New York-Heidelberg, 1976. MR 388547, DOI 10.1007/978-1-4684-6257-9
- F. Spitzer and H. Widom, The circumference of a convex polygon, Proc. Amer. Math. Soc. 12 (1961), 506–509. MR 130616, DOI 10.1090/S0002-9939-1961-0130616-7
- William F. Stout, Almost sure convergence, Probability and Mathematical Statistics, Vol. 24, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 455094
- V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 3 (1964), 211–226 (1964). MR 175194, DOI 10.1007/BF00534910
- Daniel W. Stroock, Probability theory, 2nd ed., Cambridge University Press, Cambridge, 2011. An analytic view. MR 2760872
- Paolo Tilli, Isoperimetric inequalities for convex hulls and related questions, Trans. Amer. Math. Soc. 362 (2010), no. 9, 4497–4509. MR 2645038, DOI 10.1090/S0002-9947-10-04734-3
- V. Vysotsky, The isoperimetric problem for convex hulls and the large deviations rate functionals of random walks. Preprint (2023) arXiv:2306.12359.
- Andrew R. Wade and Chang Xu, Convex hulls of planar random walks with drift, Proc. Amer. Math. Soc. 143 (2015), no. 1, 433–445. MR 3272767, DOI 10.1090/S0002-9939-2014-12239-8
- Andrew R. Wade and Chang Xu, Convex hulls of random walks and their scaling limits, Stochastic Process. Appl. 125 (2015), no. 11, 4300–4320. MR 3385604, DOI 10.1016/j.spa.2015.06.008
- V. A. Zalgaller, Extremal problems on the convex hull of a space curve, Algebra i Analiz 8 (1996), no. 3, 1–13 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 3, 369–379. MR 1402285
Bibliographic Information
- Wojciech Cygan
- Affiliation: University of Wrocław, Faculty of Mathematics and Computer Science, Institute of Mathematics, pl. Grunwaldzki 2/4, 50–384 Wrocław, Poland; and Technische Universität Dresden, Faculty of Mathematics, Institute of Mathematical Stochastics, Zellescher Weg 25, 01069 Dresden, Germany
- MR Author ID: 1095836
- ORCID: 0000-0001-7734-8644
- Email: wojciech.cygan@uwr.edu.pl
- Nikola Sandrić
- Affiliation: Department of Mathematics, University of Zagreb, Zagreb, Croatia
- Email: nsandric@math.hr
- Stjepan Šebek
- Affiliation: Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb, Croatia
- ORCID: 0000-0002-1802-1542
- Email: stjepan.sebek@fer.hr
- Andrew Wade
- Affiliation: Department of Mathematical Sciences, Durham University, Durham, United Kingdom
- MR Author ID: 741020
- ORCID: 0000-0002-3829-3406
- Email: andrew.wade@durham.ac.uk
- Received by editor(s): July 25, 2023
- Received by editor(s) in revised form: March 29, 2024
- Published electronically: July 16, 2024
- Additional Notes: This work was supported by Alexander-von-Humboldt Foundation project No. HRV 1151902 HFST-E. The work of the second and third authors was supported by Croatian Science Foundation grant no. 2277. The work of the fourth author was supported by EPSRC grant EP/W00657X/1.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 6695-6724
- MSC (2020): Primary 60G50; Secondary 60D05, 60F15, 60J65, 52A22
- DOI: https://doi.org/10.1090/tran/9238