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Random walk on half-plane half-comb structure
Random walk on half-plane half-comb structure
We study limiting properties of a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We give strong approximations for the components with random time changed Wiener processes. As consequences, limiting distributions and some laws of the iterated logarithm are presented. Finally, a formula is given for the probability that the random walk returns to the origin in 2N steps.
- Technische Universat Wien Austria
- Hungarian Academy of Science Hungary
- City University of New York United States
- Carleton University Canada
- University of Debrecen Hungary
2213
[1] Bertacchi, D. (2006). Asymptotic behaviour of the simple random walk on the 2- dimensional comb. Electron. J. Probab. 11:1184-1203. [OpenAIRE]
[2] Bertacchi, D. and Zucca, F. (2003). Uniform asymptotic estimates of transition probabilities on combs. J. Aust. Math. Soc. 75:325-353. [OpenAIRE]
[3] Borodin, A.N. and Salminen, P. (1996). Handbook of Brownian Motion - Facts and Formulae. Birkhäuser, Boston.
[4] Csáki, E., Csörgő, M., Földes, A. and Révész, P. (2009). Strong limit theorems for a simple random walk on the 2-dimensional comb. Electr. J. Probab. 14:2371-2390. [OpenAIRE]
[5] Csáki, E., Csörgő, M., Földes, A. and Révész, P. (2011). On the local time of random walk on the 2-dimensional comb. Stoch. Process. Appl. 121:1290-1314. [OpenAIRE]
[6] Csáki, E. and Grill, K. (1988). On the large values of the Wiener process. Stoch. Process. Appl. 27:43-56.
[7] Csörgő, M. and Révész, P. (1981). Strong Approximations in Probability and Statistics. Academic Press, New York.
[8] Dvoretzky, A. and Erdős, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Symposium, pp. 353-367. [OpenAIRE]
[9] Erdős, P. and Taylor, S.J. (1960). Some problems concerning the structure of random walk paths. Acta Math. Acad. Sci. Hungar. 11:137-162.
[10] Gradshteyn, I.S. and Ryzhik, I.M. (1994). Table of integrals, series, and products. Academic Press, Boston, MA.
- Technische Universat Wien Austria
- Hungarian Academy of Science Hungary
- City University of New York United States
- Carleton University Canada
- University of Debrecen Hungary
- Vienna University of Technology (TU Wien) Austria
- College of Staten Island United States
- Eszterhazy Karoly University Hungary
- Alfréd Rényi Institute of Mathematics
- Hungarian Academy of Sciences
- TU Wien Austria
- Hungarian Academy of Sciences Hungary
- Alfréd Rényi Institute of Mathematics Hungary
We study limiting properties of a random walk on the plane, when we have a square lattice on the upper half-plane and a comb structure on the lower half-plane, i.e., horizontal lines below the x-axis are removed. We give strong approximations for the components with random time changed Wiener processes. As consequences, limiting distributions and some laws of the iterated logarithm are presented. Finally, a formula is given for the probability that the random walk returns to the origin in 2N steps.