## Bernoulli

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

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

Sami Mustapha

#### 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

#### 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