Tail Asymptotics of the Occupation Measure for a Markov Additive Process with an M/G/1-Type Background Process

We are concerned with a discrete-time Markov additive process (MAP) generated by a Markov chain with transition probabilities similar to that for the M/G/1 queue. We are interested in its occupation measure before the additive component returns to the origin, and we study its asymptotic behavior as the additive component goes to infinity. This asymptotic problem is motivated by studies on the tail asymptotics of the stationary distribution of a reflected two-dimensional random walk and its applications in queueing theory. This is also related to sample path large deviations for the random walk with discontinuous statistics, which includes the present MAP as a special case. We study the asymptotic problem through the matrix moment generating function of the Markov additive kernel, called the Feynman-Kac transform. We find the right and left positive invariant vectors of this transform when its convergence parameter is one. Using these results, we completely characterize this MAP under exponential change of measure to be transient, null recurrent and positive recurrent. These results lead to an answer to how the occupation measure decays for each fixed background state as the additive component goes to infinity. We have a complete answer for rough asymptotics and a partial answer for exact asymptotics.