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Convex hull of Brownian motion ind-dimensions

Abstract

We supposeK(w) to be the boundary of the closed convex hull of a sample path ofZ t(w), 0 ≦t ≦ 1 of Brownian motion ind-dimensions. A combinatorial result of Baxter and Borndorff Neilson on the convex hull of a random walk, and a limiting process utilizing results of P. Levy on the continuity properties ofZ t(w) are used to show that the curvature ofK(w) is concentrated on a metrically small set.

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References

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    O. Barndorff-Nielson and G. Baxter,Combinational lemmas in n-dimensions, Trans. Amer. Math. Soc.108 (1963), 313–325.

  2. 2.

    L. Carleson,On a class of meromorphic functions and its associated exceptional sets. Thesis, University of Uppsala, (1950).

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    J. R. Kinney,The convex hull of plane Brownian motion, Ann. Math. Statist.34 (1963), 327–329.

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    P. Lévy,Processus stochastiques et mouvement Brownien, Gautier-Villars, Paris, 1948.

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Author information

Correspondence to J. R. Kinney.

Additional information

Research received support from ONR under contract No. Nonr-2587(02).

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Kinney, J.R. Convex hull of Brownian motion ind-dimensions. Israel J. Math. 4, 139–143 (1966). https://doi.org/10.1007/BF02937459

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Keywords

  • Brownian Motion
  • Random Walk
  • Convex Hull
  • Meromorphic Function
  • Sample Path