Abstract
We study the path behavior of the simple symmetric walk on some comb-type subsets of \({{\mathbb {Z}}}^2\) which are obtained from \({{\mathbb {Z}}}^2\) by removing all horizontal edges belonging to certain sets of values on the y-axis. We obtain some strong approximation results and discuss their consequences.
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Acknowledgements
The authors are indebted to Miklós Csörgő and Pál Révész for their inspiration and careful reading of our manuscript that greatly improved our presentation. We also would like to thank our referee for his/her insightful suggestions which made our presentation much nicer.
Funding
Funding was provided by Research Foundation of The City University of New York (Grant No: 61520-0049).
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A. Földes: Research supported by a PSC CUNY Grant, No. 61520-0049.
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Csáki, E., Földes, A. Random Walks on Comb-Type Subsets of \(\mathbb {Z}^2\). J Theor Probab 33, 2233–2257 (2020). https://doi.org/10.1007/s10959-019-00938-5
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DOI: https://doi.org/10.1007/s10959-019-00938-5
Keywords
- Random walk
- 2-dimensional comb
- Strong approximation
- 2-dimensional Wiener process
- Oscillating Brownian motion
- Laws of the iterated logarithm
- Iterated Brownian motion