On a semi-Markov generalization of the random walk

The semi-Markov process studied here is a generalized random walk on the non-negative integers with zero as a reflecting barrier, in which the time interval between two consecutive jumps is given an arbitrary distribution H(t). Our process is identical with the Markov chain studied by Miller 6] in the special case when H(t)=U₁(t), the Heaviside function with unit jump at t=1. By means of a Spitzer-Baxter type identity, we establish criteria for transience, positive and null recurrence, as well as conditions for exponential ergodicity. The results obtained here generalize those of 6] and some classical results in random walk theory 10].