Abstract
The basic problem considered in this paper is that of determining conditions for recurrence and transience for two dimensional irreducible Markov chains whose state space is Z + 2 =Z+xZ+. Assuming bounded jumps and a homogeneity condition Malyshev [7] obtained necessary and sufficient conditions for recurrence and transience of two dimensional random walks on the positive quadrant. Unfortunately, his hypothesis that the jumps of the Markov chain be bounded rules out for example, the Poisson arrival process. In this paper we generalise Malyshev's theorem by means of a method that makes novel use of the solution to Laplace's equation in the first quadrant satisfying an oblique derivative condition on the boundaries. This method, which allows one to replace the very restrictive boundedness condition by a moment condition and a lower boundedness condition, is of independent interest.
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