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Remarks on renewal theory for percolation processes

Published online by Cambridge University Press:  14 July 2016

R. T. Smythe
R. T. Smythe*
Affiliation:
University of Washington, Seattle
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Abstract

We extend some results of Hammersley and Welsh concerning first-passage percolation on the two-dimensional integer lattice. Our results include: (i) weak renewal theorems for the unrestricted reach processes; (ii) an L1-ergodic theorem for the unrestricted first-passage time from (0, 0) to the line X = n; and (iii) weakening of the boundedness restrictions on the underlying distribution in Hammersley and Welsh's weak renewal theorems for the cylinder reach processes.

Keywords

FIRST-PASSAGE PERCOLATIONRENEWAL THEORYSUBADDITIVE PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 13 , Issue 2 , June 1976 , pp. 290 - 300
Copyright
Copyright © Applied Probability Trust 1976 

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References

[1] Hammersley, J. M. (1966) First-passage percolation. J. R. Statist. Soc. B 28, 491–496.Google Scholar
[2] Hammersley, J. M. and Welsh, D. J. A. (1965) First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, Bernoulli, Bayes, Laplace Anniversary Volume. Springer-Verlag, Berlin, 61–110.Google Scholar
[3] Kingman, J. F. C. (1968) The ergodic theory of subadditive stochastic processes. J. R. Statist. Soc. B 30, 499–510.Google Scholar
[4] Meyer, P. A. (1966) Probability and Potentials. Blaisdell, New York.Google Scholar