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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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
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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