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Martin boundary of a reflected random walk on a half-space
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  • Published: 30 June 2009

Martin boundary of a reflected random walk on a half-space

  • Irina Ignatiouk-Robert1 

Probability Theory and Related Fields volume 148, pages 197–245 (2010)Cite this article

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Abstract

The complete representation of the Martin compactification for reflected random walks on a half-space \({\mathbb{Z}^d\times\mathbb{N}}\) is obtained. It is shown that the full Martin compactification is in general not homeomorphic to the “radial” compactification obtained by Ney and Spitzer for the homogeneous random walks in \({\mathbb{Z}^d}\): convergence of a sequence of points \({z_n\in\mathbb{Z}^{d-1}\times\mathbb{N}}\) to a point of on the Martin boundary does not imply convergence of the sequence z n /|z n | on the unit sphere S d. Our approach relies on the large deviation properties of the scaled processes and uses Pascal’s method combined with the ratio limit theorem. The existence of non-radial limits is related to non-linear optimal large deviation trajectories.

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Authors and Affiliations

  1. Département de Mathématiques, Université de Cergy-Pontoise, 2, Avenue Adolphe Chauvin, 95302, Cergy-Pontoise Cedex, France

    Irina Ignatiouk-Robert

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  1. Irina Ignatiouk-Robert
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Correspondence to Irina Ignatiouk-Robert.

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Ignatiouk-Robert, I. Martin boundary of a reflected random walk on a half-space. Probab. Theory Relat. Fields 148, 197–245 (2010). https://doi.org/10.1007/s00440-009-0228-4

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  • Received: 14 October 2006

  • Revised: 27 April 2009

  • Published: 30 June 2009

  • Issue Date: September 2010

  • DOI: https://doi.org/10.1007/s00440-009-0228-4

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Keywords

  • Martin boundary
  • Sample path large deviations
  • Random walk

Mathematics Subject Classification (2000)

  • Primary 60J50
  • Secondary 60F10
  • 60J45
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