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February 2007 Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments
Sami Mustapha
Bernoulli 13(1): 131-147 (February 2007). DOI: 10.3150/07-BEJ5135


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.


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Sami Mustapha. "Gambler’s ruin estimates for random walks with symmetric spatially inhomogeneous increments." Bernoulli 13 (1) 131 - 147, February 2007.


Published: February 2007
First available in Project Euclid: 30 March 2007

MathSciNet: MR2307398
zbMATH: 1111.62070
Digital Object Identifier: 10.3150/07-BEJ5135

Keywords: discrete potential theory , Gaussian estimates , Markov chains , transition kernels

Rights: Copyright © 2007 Bernoulli Society for Mathematical Statistics and Probability


Vol.13 • No. 1 • February 2007
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