Electronic Communications in Probability

The mean number of sites visited by a random walk pinned at a distant point

Kôhei Uchiyama

Abstract

This paper concerns the number $Z_n$ of sites visited up to time $n$ by a random walk  $S_n$ having zero mean and moving on the two dimensional square lattice ${\bf Z}^2$.    Asymptotic evaluation of  the conditional expectation of  $Z_n$ for large  $n$ given that $S_n=x$ is carried out under some exponential moment condition. It gives an explicit form of the leading term valid uniformly in $(x, n)$, $|x|< cn$ for each $c>0$.

Article information

Source
Electron. Commun. Probab., Volume 20 (2015), paper no. 17, 9 pp.

Dates
Accepted: 24 February 2015
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465320944

Digital Object Identifier
doi:10.1214/ECP.v20-4027

Mathematical Reviews number (MathSciNet)
MR3320405

Zentralblatt MATH identifier
1325.60070

Subjects

Rights

Citation

Uchiyama, Kôhei. The mean number of sites visited by a random walk pinned at a distant point. Electron. Commun. Probab. 20 (2015), paper no. 17, 9 pp. doi:10.1214/ECP.v20-4027. https://projecteuclid.org/euclid.ecp/1465320944

References

• van den Berg, M.; Bolthausen, E. On the expected volume of the Wiener sausage for a Brownian bridge. Math. Z. 224 (1997), no. 1, 33–48.
• Dvoretzky, A.; ErdÃ¶s, P. Some problems on random walk in space. Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950. pp. 353–367. University of California Press, Berkeley and Los Angeles, 1951.
• Le Gall, J.-F. Sur une conjecture de M. Kac. (French) [On a conjecture of M. Kac] Probab. Theory Related Fields 78 (1988), no. 3, 389–402.
• Hamana, Yuji. A remark on the range of three dimensional pinned random walks. Kumamoto J. Math. 19 (2006), 83–97.
• Kingman, J. F. C. Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. London Math. Soc. (3) 13 1963 593–604.
• McGillivray, I. The spectral shift function for planar obstacle scattering at low energy. Math. Nachr. 286 (2013), no. 11-12, 1208–1239.
• McGillivray, I. Large time volume of the pinned Wiener sausage to second order. Math. Nachr. 284 (2011), no. 2-3, 142–165.
• W. Feller, An Introduction to Probability Theory and Its Applications, vol. 1.
• Gnedenko, B. V.; Kolmogorov, A. N. Limit distributions for sums of independent random variables. Translated and annotated by K. L. Chung. With an Appendix by J. L. Doob. Addison-Wesley Publishing Company, Inc., Cambridge, Mass., 1954. ix+264 pp.
• 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.
• Spitzer, Frank. Some theorems concerning $2$-dimensional Brownian motion. Trans. Amer. Math. Soc. 87 1958 187–197.
• Uchiyama, Kôhei. The mean number of sites visited by a pinned random walk. Math. Z. 261 (2009), no. 2, 277–295.
• Uchiyama, Kôhei. The first hitting time of a single point for random walks. Electron. J. Probab. 16 (2011), no. 71, 1960–2000.
• Uchiyama, Kôhei. The expected area of the Wiener sausage swept by a disc. Stochastic Process. Appl. 123 (2013), no. 1, 191–211.