Probability Theory and Related Fields

, Volume 150, Issue 3, pp 373-403

First online:

Number of distinct sites visited by a random walk with internal states

  • Péter NándoriAffiliated withInstitute of Mathematics, Budapest University of Technology and Economics Email author 

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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.


Random walk with internal states Markov chain Visited points Local limit theorem Law of large numbers

Mathematics Subject Classification (2000)

60J10 82C41