Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Cut-set sums and tree processes


Author: K. J. Falconer
Journal: Proc. Amer. Math. Soc. 101 (1987), 337-346
MSC: Primary 90B10; Secondary 60G48, 60J80, 90B15
DOI: https://doi.org/10.1090/S0002-9939-1987-0902553-7
MathSciNet review: 902553
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose that an infinite tree has a value assigned to each vertex. We obtain estimates for the sums of such values over cut-sets of the tree. For certain tree processes, where the values are given by random variables, we investigate the almost sure behavior of such cut-set sums. Processes of this type arise in problems concerning random fractals and flows in random networks.


References [Enhancements On Off] (What's this?)

  • [1] K. B. Athreya and P. E. Ney, Branching processes, Springer-Verlag, Berlin and New York, 1972. MR 0373040 (51:9242)
  • [2] T. Bedford, Dimension and dynamics for fractal recurrent sets, J. London Math. Soc. 33 (1986), 89-100. MR 829390 (87g:28004)
  • [3] L. Egghe, Stopping time techniques for analysts and prohabilists, Cambridge Univ. Press, 1984. MR 808582 (87j:60070)
  • [4] K. Falconer, The geometry of fractal sets, Cambridge Univ. Press, 1985. MR 867284 (88d:28001)
  • [5] -, Random fractals, Math. Proc. Cambridge Philos. Soc. 100 (1986), 559-582. MR 857731 (88e:28005)
  • [6] S. Graf, Statistically self-similar fractals, Prob. Theory Related Fields 74 (1987), 357-392. MR 873885 (88c:60038)
  • [7] G. R. Grimett, Random flows: network flows and electrical flows through random media, Surveys in Combinatorics, Cambridge Univ. Press, 1985, pp. 59-95. MR 822770 (87i:90104)
  • [8] K. Krickeberg, Convergence of martingales with a directed index set, Trans. Amer. Math. Soc. 83 (1956), 313-337. MR 0091328 (19:947e)
  • [9] R. D. Mauldin and S. C. Williams, Random constructions, asymptotic geometric and topological properties, Trans. Amer. Math. Soc. 295 (1986), 325-346. MR 831202 (87j:60027)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 90B10, 60G48, 60J80, 90B15

Retrieve articles in all journals with MSC: 90B10, 60G48, 60J80, 90B15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0902553-7
Keywords: Tree process, network flow, branching process, random fractal
Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society