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Random walks in the high-dimensional limit II: The crinkled subordinator

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  • Kabluchko, Zakhar
  • Marynych, Alexander
  • Raschel, Kilian

Abstract

A crinkled subordinator is an ℓ2-valued random process which can be thought of as a version of the usual one-dimensional subordinator with each out of countably many jumps being in a direction orthogonal to the directions of all other jumps. We show that the path of a d-dimensional random walk with n independent identically distributed steps with heavy-tailed distribution of the radial components and asymptotically orthogonal angular components converges in distribution in the Hausdorff distance up to isometry and also in the Gromov–Hausdorff sense, if viewed as a random metric space, to the closed range of a crinkled subordinator, as d,n→∞.

Suggested Citation

  • Kabluchko, Zakhar & Marynych, Alexander & Raschel, Kilian, 2024. "Random walks in the high-dimensional limit II: The crinkled subordinator," Stochastic Processes and their Applications, Elsevier, vol. 176(C).
    • Handle: RePEc:eee:spapps:v:176:y:2024:i:c:s0304414924001340
    DOI: 10.1016/j.spa.2024.104428
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