The hitting distributions of a line for two dimensional random walks
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Abstract:
For every irreducible random walk on $\mathbf {Z}^2$ with zero mean and finite $2+\delta$ absolute moment ($0\leq \delta <1$) we obtain fine asymptotic estimates of the probability that the first visit of the walk to the horizontal axis takes place at a specified site of it.References
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- Yasunari Fukai and Kôhei Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24 (1996), no. 4, 1979–1992. MR 1415236, DOI 10.1214/aop/1041903213
- Wassily Hoeffding, On sequences of sums of independent random vectors, Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II, Univ. California Press, Berkeley, Calif., 1961, pp. 213–226. MR 0138116
- Takahiro Kazami and Kôhei Uchiyama, Random walks on periodic graphs, Trans. Amer. Math. Soc. 360 (2008), no. 11, 6065–6087. MR 2425703, DOI 10.1090/S0002-9947-08-04451-6
- Harry Kesten, Hitting probabilities of random walks on $\textbf {Z}^d$, Stochastic Process. Appl. 25 (1987), no. 2, 165–184. MR 915132, DOI 10.1016/0304-4149(87)90196-7
- Gregory F. Lawler and Vlada Limic, The Beurling estimate for a class of random walks, Electron. J. Probab. 9 (2004), no. 27, 846–861. MR 2110020, DOI 10.1214/EJP.v9-228
- Frank Spitzer, Principles of random walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1964. MR 0171290, DOI 10.1007/978-1-4757-4229-9
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Kôhei Uchiyama, Green’s functions for random walks on $\textbf {Z}^N$, Proc. London Math. Soc. (3) 77 (1998), no. 1, 215–240. MR 1625467, DOI 10.1112/S0024611598000458
- Kôhei Uchiyama, Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes, Electron. J. Probab. 12 (2007), no. 6, 138–180. MR 2299915, DOI 10.1214/EJP.v12-396
- K. Uchiyama, The hitting distributions of a half real line for two dimensional random walks. to appear in Arkiv för Mat.
- K. Uchiyama, The hitting time of a single point for random walks. (2008), http://www.math.titech.ac.jp/ tosho/Preprints/index-j.html.
- K. Uchiyama, The hitting distribution of line segments for two dimensional random walks. (in preparation)
- K. Uchiyama, One dimensional lattice random walks killed at a point or on a line. (preprint)
- K. Uchiyama, The random walks on the upper half plane. preprint.
- A. Zygmund, Trigonometric series. Vol. I, II, 3rd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002. With a foreword by Robert A. Fefferman. MR 1963498
Additional Information
- Kôhei Uchiyama
- Affiliation: Department of Mathematics, Tokyo Institute of Technology, Oh-okayama, Meguro Tokyo 152-8551, Japan
- Email: uchiyama@math.titech.ac.jp
- Received by editor(s): March 28, 2008
- Published electronically: December 3, 2009
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 2559-2588
- MSC (2010): Primary 60G50; Secondary 60J45
- DOI: https://doi.org/10.1090/S0002-9947-09-05072-7
- MathSciNet review: 2584611