Asymptotic properties of the occupation measure in a multidimensional skip-free Markov-modulated random walk

We consider a discrete-time d-dimensional process {X-n} = {(X-1,X-n, X-2,X-n,..., X-d,X-n)} on Z(d) with a background process {J(n)} on a countable set S-0, where individual processes {X-i,X-n}, i is an element of {1, 2,..., d}, are skip free. We assume that the joint process {Y-n} = {( Xn, Jn)} is Markovian and that the transition probabilities of the d-dimensional process {X-n} vary according to the state of the background process {J(n)}. This modulation is assumed to be space homogeneous. We refer to this process as a d-dimensional skip-free Markov-modulated random walk. For y, y' is an element of Z(+)(d) x S-0, consider the process {Y-n}(n >= 0) starting from the state y and let ((q) over tilde (y,y') be the expected number of visits to the state y' before the process leaves the nonnegative area Z(+)(d) x S-0 for the first time. For y = (x, j) is an element of Z(+)(d) x S-0, the measure ((q) over tilde (y, y'); y' = (x', j') is an element of Z(+)(d) x S-0) is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupationmeasure as x' goes to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measure.