Bernoulli

  • Bernoulli
  • Volume 13, Number 1 (2007), 131-147.

Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments

Sami Mustapha

Full-text: Open access

Abstract

Generalizing to higher dimensions the classical gambler’s ruin estimates, we give pointwise estimates for the transition kernel corresponding to a spatially inhomogeneous random walk on the half-space. Our results hold under some strong but natural assumptions of symmetry, boundedness of the increments, and ellipticity. Among the most important steps in our proof are: discrete variants of the boundary Harnack estimate, as proven by Bauman, Bass and Burdzy, and Fabes et al., based on comparison arguments and potential-theoretical tools; the existence of a positive $\tilde{L}$-harmonic function globally defined in the half-space; and some Gaussian inequalities obtained by a treatment inspired by Varopoulos.

Article information

Source
Bernoulli, Volume 13, Number 1 (2007), 131-147.

Dates
First available in Project Euclid: 30 March 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1175287724

Digital Object Identifier
doi:10.3150/07-BEJ5135

Mathematical Reviews number (MathSciNet)
MR2307398

Zentralblatt MATH identifier
1111.62070

Keywords
discrete potential theory Gaussian estimates Markov chains transition kernels

Citation

Mustapha, Sami. Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments. Bernoulli 13 (2007), no. 1, 131--147. doi:10.3150/07-BEJ5135. https://projecteuclid.org/euclid.bj/1175287724


Export citation