The First Hitting Time of a Single Point for Random Walks

Kohei Uchiyama (Tokyo Institute of Technology)

Abstract


This paper concerns the first hitting time $T_0$ of the origin for random walks on $d$-dimensional integer lattice with zero mean and a finite $2+\delta$ absolute moment ($\delta\geq0$). We derive detailed asymptotic estimates of the probabilities $\mathbb{P}_x(T_0=n)$ as $n\to\infty$ that are valid uniformly in $x$, the position at which the random walks start.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1960-2000

Publication Date: October 21, 2011

DOI: 10.1214/EJP.v16-931

References

  1. R. Durrett and D. Remenik, Voter model perturbations in two dimensions (private communication)
  2. Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F. G. Tables of integral transforms. Vol. I.Based, in part, on notes left by Harry Bateman.McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954. xx+391 pp. MR0061695 (15,868a)
  3. Feller, William. An introduction to probability theory and its applications. Vol. I.2nd ed.John Wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London, 1957. xv+461 pp. MR0088081 (19,466a)
  4. Fukai, Yasunari; Uchiyama, Kôhei. Potential kernel for two-dimensional random walk. Ann. Probab. 24 (1996), no. 4, 1979--1992. MR1415236 (97m:60098)
  5. Hoeffding, Wassily. On sequences of sums of independent random vectors. 1961 Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. II pp. 213--226 Univ. California Press, Berkeley, Calif. MR0138116 (25 #1563)
  6. Hamana, Yuji. A remark on the range of three dimensional pinned random walks. Kumamoto J. Math. 19 (2006), 83--97. MR2211634 (2006k:60080)
  7. Hamana, Yuji. On the range of pinned random walks. Tohoku Math. J. (2) 58 (2006), no. 3, 329--357. MR2273274 (2008e:60128)
  8. Hardy, G. H. Ramanujan. Twelve lectures on subjects suggested by his life and work.Cambridge University Press, Cambridge, England; Macmillan Company, New York, 1940. vii+236 pp. MR0004860 (3,71d)
  9. Jain, Naresh C.; Pruitt, William E. The range of random walk. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, pp. 31--50. Univ. California Press, Berkeley, Calif., 1972. MR0410936 (53 #14677)
  10. Kesten, Harry. Ratio theorems for random walks. II. J. Analyse Math. 11 1963 323--379. MR0163365 (29 #668)
  11. Spitzer, Frank. Principles of random walk.The University Series in Higher Mathematics D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964 xi+406 pp. MR0171290 (30 #1521)
  12. Uchiyama, Kôhei. Green's functions for random walks on ${\bf Z}\sp N$. Proc. London Math. Soc. (3) 77 (1998), no. 1, 215--240. MR1625467 (99f:60132)
  13. Uchiyama, Kôhei. The hitting distributions of a line for two dimensional random walks. Trans. Amer. Math. Soc. 362 (2010), no. 5, 2559--2588. MR2584611 (2011b:60173)
  14. Uchiyama, Kôhei. Asymptotic estimates of the Green functions and transition probabilities for Markov additive processes. Electron. J. Probab. 12 (2007), no. 6, 138--180. MR2299915 (2008d:60092)
  15. Uchiyama, Kôhei. The mean number of sites visited by a pinned random walk. Math. Z. 261 (2009), no. 2, 277--295. MR2457300 (2010e:60096)
  16. Uchiyama, Kôhei. One dimensional lattice random walks with absorption at a point/on a half line. J. Math. Soc. Japan 63 (2011), no. 2, 675--713. MR2793114
  17. K. Uchiyama, Asymptotic estimates of the distribution of Brownian hitting time of a disc., to appear in J. Theor. Probab., Available at ArXiv:1007.4633xl [math.PR].
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.