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Asymptotics for weighted minimal spanning trees on random points

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  • Yukich, J. E.

Abstract

For all p[greater-or-equal, slanted]1 let Mp(X1,...,Xn) denote the length of the minimal spanning tree through random variables X1,...,Xn, where the cost of an edge (Xi, Xj) is given by Xi-Xjp. If the Xi, i[greater-or-equal, slanted]1, are i.i.d. with values in [0,1]d, d[greater-or-equal, slanted]2, and have a density f which is bounded away from zero and which has support [0,1]d, then for all p[greater-or-equal, slanted]1, including p in the critical range p[greater-or-equal, slanted]d, we haveHere C(p,d) denotes a positive constant depending only on p and d and c.c. denotes complete convergence. Extensions to related optimization problems are indicated and rates of convergence are also found.

Suggested Citation

  • Yukich, J. E., 2000. "Asymptotics for weighted minimal spanning trees on random points," Stochastic Processes and their Applications, Elsevier, vol. 85(1), pages 123-138, January.
    • Handle: RePEc:eee:spapps:v:85:y:2000:i:1:p:123-138
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    References listed on IDEAS

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    1. Redmond, C. & Yukich, J. E., 1996. "Asymptotics for Euclidean functionals with power-weighted edges," Stochastic Processes and their Applications, Elsevier, vol. 61(2), pages 289-304, February.
    2. Lee, Sungchul, 1999. "Asymptotics of power-weighted Euclidean functionals," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 109-116, January.
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    Cited by:

    1. Lee, Sungchul, 2000. "Rate of convergence of power-weighted Euclidean minimal spanning trees," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 163-176, March.

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