Abstract
For every two-dimensional random walk on the square lattice Z 2 having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s,0), when it is started at a site far from both (0,s) and the origin.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bousquet-Mélou, M. and Schaeffer, G., Walks on the slit plane, Probab. Theory Related Fields 124 (2002), 305–344.
Fukai, Y., Hitting distribution to a quadrant of two-dimensional random walk, Kodai Math. J. 23 (2000), 35–80.
Fukai, Y., Hitting time of a half-line by two-dimensional random walk, Probab. Theory Related Fields 128 (2004), 323–346.
Kesten, H., Hitting probabilities of random walks on Z d, Stochastic Process. Appl. 25 (1987), 165–184.
Kesten, H., How long are the arms of DLA, J. Phys. A 20 (1987), L29–L33.
Kesten, H., Some caricatures of the multiple contact diffusion-limitted aggregation and η-model, in Stochastic Analysis (Durham, 1991 ), pp. 179–227, Cambridge Univ. Press, Cambridge, 1991.
Lawler, G. F. and Limic, V., The Beurling estimate for a class of random walks, Electron. J. Probab. 9 (2004), 846–861.
Rosén, B., On the asymptotic distribution of sums of independent identically distributed random variables, Ark. Mat. 4 (1962), 323–332.
Rudin, W., Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
Spitzer, F., Principles of Random Walks, Van Nostrand, Princeton, NJ, 1964.
Uchiyama, K., Green’s functions for random walks on Z d, Proc. London Math. Soc. 77 (1998), 215–240.
Uchiyama, K., The hitting distributions of a line for two dimensional random walks, to appear in Trans. Amer. Math. Soc.
Uchiyama, K., The hitting distribution of line segments for two dimensional random walks, Preprint, 2008.
Yosida, K., Functional Analysis, 6th ed., Springer, Berlin–Heidelberg, 1980.
Zygmund, A., Trigonometric Series, vol. 2, 2nd ed., Cambridge Univ. Press, Cambridge, 1959.
Author information
Authors and Affiliations
Corresponding author
Additional information
An erratum to this article can be found at http://dx.doi.org/10.1007/s11512-011-0162-4
Rights and permissions
About this article
Cite this article
Uchiyama, K. The hitting distributions of a half real line for two-dimensional random walks. Ark Mat 48, 371–393 (2010). https://doi.org/10.1007/s11512-009-0096-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11512-009-0096-2