Abstract
In the classical paper of Dvoretzky and Erdős (Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp 353–367, 1951), asymptotics for the expected value and the variance of the number of distinct sites visited by a Simple Symmetric Random Walk were calculated. Here, these results are generalized for Random Walks with Internal States. Moreover, both weak and strong laws of large numbers are proved. As a tool for these results, the error term of the local limit theorem in Krámli and Szász (Zeitschrift Wahrscheinlichkeitstheorie verw Gebiete 63:85–95, 1983) is also estimated.
Article PDF
Similar content being viewed by others
References
Bass R.F., Rosen J.: An almost sure invariance principle for the range of random walks. Ann. Probab. 33, 1856–1885 (2005)
Bender, E.A., Lawler, G.F., Pemantle, R., Wilf, H.S.: Irreducible compositions and the first return to the origin of a random walk. Seminaire Lotharingien de Combinatoire 50, B50h, pp. 13 (2004)
Chen X.: Moderate and small deviations for the ranges of one-dimensional random walks. J. Theor. Probab. 19, 721–739 (2006)
Dvoretzky, A., Erdős, P.: Some problems on random walk in space. Proceedings of the 2nd Berkeley Symposium on Mathematical Statistics and Probability, pp. 353–367 (1951)
Feller W.: An introduction to Probability Theory and Its Applications, 2nd edn, vol. 2. John Wiley and Sons, New York (1971)
Hughes B.D.: Random Walks and Random Environments: Random walks. Oxford University Press, Oxford (1995)
Ibragimov I.A., Linnik Yu.V.: Independent and Stationarily Depending Variables (in Russian). Nauka, Moscow (1965)
Jain N.C., Pruitt W.E.: The Range of Recurrent Random Walk in the Plane. Zeitschrift Wahrscheinlichkeitstheorie verw. Gebiete 16, 279–292 (1970)
Kato T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980)
Krámli A., Szász D.: Random walks with internal degrees of freedom I. Zeitschrift Wahrscheinlichkeitstheorie verw. Gebiete 63, 85–95 (1983)
Krámli A., Szász D.: Random walks with internal degrees of freedom II. Zeitschrift Wahrscheinlichkeitstheorie verw. Gebiete 68, 53–64 (1984)
Krámli A., Simányi N., Szász D.: Random walks with internal degrees of freedom III. Zeitschrift Wahrscheinlichkeitstheorie verw. Gebiete 72, 603–617 (1986)
Petrov V.V.: Sums of Independent Random Variables. Akademie Verlag, Berlin (1975)
Pène F.: Planar Lorentz gas in a random scenery. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 45, 818–839 (2009)
Pène F.: Asymptotic of the number of obstacles visited by a planar Lorentz process. Discrete Cont. Dyn. Syst. Ser. A 24, 567–588 (2009)
Sinai, Ya.G.: Random walks and some problems concerning Lorentz gas. Proceedings of the Kyoto Conference, pp. 6–17 (1981)
Telcs A.: Random walks with internal states (in Hungarian). MTA Sztaki Tanulmányok 145, 1–60 (1983)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nándori, P. Number of distinct sites visited by a random walk with internal states. Probab. Theory Relat. Fields 150, 373–403 (2011). https://doi.org/10.1007/s00440-010-0277-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-010-0277-8