Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The hitting distributions of a line for two dimensional random walks


Author: Kôhei Uchiyama
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
Published electronically: December 3, 2009
MathSciNet review: 2584611
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. W. Feller, An Introduction to Probability Theory and Its Applications, vol. 2, John Wiley and Sons, Inc. (1966). MR 0210154 (35:1048)
  • 2. Y. Fukai and K. Uchiyama, Potential kernel for two-dimensional random walk, Ann. Probab. 24 (1992) 1979-1992. MR 1415236 (97m:60098)
  • 3. W. Hoeffding, On sequences of sums of independent random vectors, Fourth Berkeley Symp. on Stat. and Probab., vol. II (1961), 213-226. MR 0138116 (25:1563)
  • 4. T. Kazami and K. Uchiyama, Random walks on periodic graphs, Trans. Amer. Math. Soc. 360 (2008) 6065-6087. MR 2425703
  • 5. H. Kesten, Hitting probabilities of random walks on $ \mathbf{Z}^d$, Stoch. Proc. Appl. 25 (1987) 165-184. MR 915132 (89a:60163)
  • 6. G. F. Lawler and V. Limic, The Beurling estimate for a class of random walks, Elec. Jour. Probab. 9 (2004) 846-861. MR 2110020 (2005k:60143)
  • 7. F. Spitzer, Principles of Random Walks, Van Nostrand, Princeton, NJ, 1964. MR 0171290 (30:1521)
  • 8. E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. MR 0304972 (46:4102)
  • 9. K. Uchiyama, Green's functions for random walks on $ \mathbf{Z}^N$, Proc. London Math. Soc. 77 (1998), 215-240. MR 1625467 (99f:60132)
  • 10. K. Uchiyama, Asymptotic estimates of Green's functions and transition probabilities for Markov additive processes, Elec. J. Probab. 12 (2007), 138-180. MR 2299915 (2008d:60092)
  • 11. K. Uchiyama, The hitting distributions of a half real line for two dimensional random walks. to appear in Arkiv för Mat.
  • 12. K. Uchiyama, The hitting time of a single point for random walks. (2008), http://www.math.titech.ac.jp/ tosho/Preprints/index-j.html.
  • 13. K. Uchiyama, The hitting distribution of line segments for two dimensional random walks. (in preparation)
  • 14. K. Uchiyama, One dimensional lattice random walks killed at a point or on a line. (preprint)
  • 15. K. Uchiyama, The random walks on the upper half plane. preprint.
  • 16. A. Zygmund, Trigonometric series, vol. 1, 3rd ed., Cambridge Univ. Press (2002) MR 1963498 (2004h:01041)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 60G50, 60J45

Retrieve articles in all journals with MSC (2010): 60G50, 60J45


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

DOI: https://doi.org/10.1090/S0002-9947-09-05072-7
Keywords: First visited site, asymptotic formula, Fourier analysis, random walk of zero mean and finite variances
Received by editor(s): March 28, 2008
Published electronically: December 3, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society