Skip to main content Accessibility help
×
×
Home

Optimal reward on a sparse tree with random edge weights

  • Davar Khoshnevisan (a1) and Thomas M. Lewis (a2)

Abstract

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.

Copyright

Corresponding author

Postal address: Department of Mathematics, University of Utah, Salt Lake City, UT 84112–0090, USA.
∗∗ Postal address: Department of Mathematics, Furman University, Greenville, SC 29613, USA. Email address: tom.lewis@furman.edu

References

Hide All
Aldous, D. (1992). Greedy search on a tree with random edge-weights. Comb. Prob. Comput. 1, 281293.
Benjamini, I., and Peres, Y. (1994). Tree-indexed random walks and first-passage percolation. Prob. Theory Relat. Fields 98, 91112.
Biggins, J. D. (1977). Chernoff's theorem in the branching random walk. J. Appl. Prob. 14, 630636.
Chung, K. L. (1974). A First Course in Probability Theory, 2nd edn. Academic Press, New York.
De Acosta, A. (1983). A new proof of the Hartman—Wintner law of the iterated logarithm. Ann. Prob. 11, 270276.
De Acosta, A., and Kuelbs, J. (1981). Some results on the cluster set C({Sn/an}) and the LIL. Ann. Prob. 11, 102122.
Dembo, A., and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd edn. Springer, Berlin.
Dubins, L., and Freedman, D. (1967). Random distribution functions. In Proc. Fifth Berkeley Symp. Math. Statist. Prob., Vol. II, eds Le Cam, L. and Neyman, J., University of California Press, Berkeley, CA, pp. 183214.
Esary, J. D., Proschan, F., and Walkup, D. W. (1967). Association of random variables, with applications Ann. Math. Statist. 38, 14661474.
Hammersley, J. M. (1974). Postulates for subadditive processes. Ann. Prob. 2, 652680.
Kingman, J. F. C. (1975). The birth problem for an age-dependent branch process. Ann. Prob. 12, 341345.
Lyons, R., and Pemantle, R. (1992). Random walks in a random environment and first-passage percolation on trees. Ann. Prob. 20, 125136.
Peres, Y. (1999). Probability on trees: an introductory climb. In Lectures on Probability Theory and Statistics (Lecture Notes Math. 1717), ed. Bernard, P., Berlin, Springer, pp. 193280.
Virág, B. (2002). Fast graphs for the random walker. Prob. Theory Relat. Fields 124, 5074.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed