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Invariance principle for the capacity and the cardinality of the range of stable random walks

Author

Listed:
  • Cygan, Wojciech
  • Sandrić, Nikola
  • Šebek, Stjepan

Abstract

We prove an almost sure invariance principle for the capacity and the cardinality of the range of a class of α-stable random walks on the integer lattice Zd with d/α>5/2, and d/α>3/2, respectively. As a direct consequence, we conclude Khintchine’s and Chung’s laws of the iterated logarithm for both processes.

Suggested Citation

  • Cygan, Wojciech & Sandrić, Nikola & Šebek, Stjepan, 2023. "Invariance principle for the capacity and the cardinality of the range of stable random walks," Stochastic Processes and their Applications, Elsevier, vol. 163(C), pages 61-84.
    • Handle: RePEc:eee:spapps:v:163:y:2023:i:c:p:61-84
    DOI: 10.1016/j.spa.2023.05.012
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    References listed on IDEAS

    as
    1. Hamana, Yuji, 1998. "An almost sure invariance principle for the range of random walks," Stochastic Processes and their Applications, Elsevier, vol. 78(2), pages 131-143, November.
    2. Xia Chen, 2006. "Moderate and Small Deviations for the Ranges of One-Dimensional Random Walks," Journal of Theoretical Probability, Springer, vol. 19(3), pages 721-739, December.
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