Martin boundary of a killed random walk on $\Z_+^d$

Details Martin boundary of a killed random walk on $\Z_+^d$ Martin boundary of a killed random walk on $\Z_+^d$

2009-09-22

arxiv.org/abs/0909.3921v1

Mathematics

Probability

Scientific paper

The Martin compactification is investigated for a d-dimensional random walk which is killed when at least one of it's coordinates becomes zero or negative. The limits of the Martin kernel are represented in terms of the harmonic functions of the associated induced Markov chains. It is shown that any sequence of points x_n with lim_n |x_n| = \infty and lim_n x_n/|x_n|= q is fundamental in the Martin compactification if up to the multiplication by constants, the induced Markov chain corresponding to the direction q has a unique positive harmonic function. The full Martin compactification is obtained for Cartesian products of one-dimensional random walks. The methods involve a ratio limit theorem and a large deviation principle for sample paths of scaled processes leading to the logarithmic asymptotics of the Green function.

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