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An almost sure invariance principle for the range of random walks

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  • Hamana, Yuji

Abstract

The range of random walks means the number of distinct sites visited at least once by the random walk before time n. We study an almost sure invariance principle for the range of random walks on the four or more dimensional integer lattice and obtain that the centralized and linearly interpolated range of the random walk can be asymptotically equal to a Brownian motion almost surely.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:78:y:1998:i:2:p:131-143
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    Cited by:

    1. 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.
    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.
    3. Kazuki Okamura, 2021. "Some Results for Range of Random Walk on Graph with Spectral Dimension Two," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1653-1688, September.
    4. Yuji Hamana, 2001. "Asymptotics of the Moment Generating Function for the Range of Random Walks," Journal of Theoretical Probability, Springer, vol. 14(1), pages 189-197, January.
    5. Mikhail Menshikov & Serguei Popov, 2014. "On Range and Local Time of Many-dimensional Submartingales," Journal of Theoretical Probability, Springer, vol. 27(2), pages 601-617, June.

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