Electronic Communications in Probability

The greedy walk on an inhomogeneous Poisson process

Katja Gabrysch and Erik Thörnblad

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The greedy walk is a deterministic walk that always moves from its current position to the nearest not yet visited point. In this paper we consider the greedy walk on an inhomogeneous Poisson point process on the real line. We prove that the property of visiting all points of the point process satisfies a $0$–$1$ law and determine explicit sufficient and necessary conditions on the mean measure of the point process for this to happen. Moreover, we provide precise results on threshold functions for the property of visiting all points.

Article information

Electron. Commun. Probab. Volume 23 (2018), paper no. 14, 11 pp.

Received: 19 December 2016
Accepted: 14 February 2018
First available in Project Euclid: 27 February 2018

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Digital Object Identifier

Primary: 60K37: Processes in random environments
Secondary: 60G55: Point processes 60K25: Queueing theory [See also 68M20, 90B22]

greedy walk inhomogeneous Poisson point processes threshold

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Gabrysch, Katja; Thörnblad, Erik. The greedy walk on an inhomogeneous Poisson process. Electron. Commun. Probab. 23 (2018), paper no. 14, 11 pp. doi:10.1214/18-ECP119. https://projecteuclid.org/euclid.ecp/1519722244

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  • [1] Bingham, N. H., Goldie, C. M. and Teugels, J. L. Regular Variation. Cambridge University Press, Cambridge, 1989.
  • [2] Bordenave, C., Foss, S. and Last, G. On the greedy walk problem. Queueing Syst. 68, (2011), 333–338.
  • [3] Foss, S., Rolla, L. T. and Sidoravicius, V. Greedy walk on the real line. Ann. Prob. 43, (2015), 1399–1418.
  • [4] Gabrysch, K. Distribution of the smallest visited point in a greedy walk on the line. J. Appl. Prob. 53, (2016), 880–887.
  • [5] Kallenberg, O. Foundations of Modern Probability. Springer-Verlag, New York, 1997.
  • [6] Kingman, J. F. C. Poisson Processes. Oxford University Press, New York, 1993.
  • [7] Rolla, L. T., Sidoravicius, V. and Tournier, L. Greedy clearing of persistent Poissonian dust. Stochastic Process. Appl. 124, (2014), 3496–3506.