We consider a discrete-time $d$-dimensional process $\{ \boldsymbol{ X}_n\}=\{ (X_{ 1,n},X_{ 2,n},...,X_{ d,n})\}$ on $\mathbb{ Z}^d$ with a background process $\{ J_n\}$ on a countable set $S_0$, where individual processes $\{ X_{ i,n}\},i\in\{ 1,2,...,d\},$ are skip free. We assume that the joint process $\{ \boldsymbol{ Y}_n\}=\{ (\boldsymbol{ X}_n,J_n)\}$ is Markovian and that the transition probabilities of the $d$-dimensional process $\{ \boldsymbol{ X}_n\}$ 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 modulate random walk. For $\boldsymbol{ y}, \boldsymbol{ y}'\in \mathbb{ Z}_+^d\times S_0$, consider the process $\{ \boldsymbol{ Y}_n\}_{ n\ge 0}$ starting from the state $\boldsymbol{ y}$ and let $\tilde{ q}_{ \boldsymbol{ y},\boldsymbol{ y}'}$ be the expected number of visits to the state $\boldsymbol{ y}'$ before the process leaves the nonnegative area $\mathbb{ Z}_+^d\times S_0$ for the first time. For $\boldsymbol{ y}=(\boldsymbol{ x},j)\in \mathbb{ Z}_+^d\times S_0$, the measure $(\tilde{ q}_{ \boldsymbol{ y},\boldsymbol{ y}'}; \boldsymbol{ y}'=(\boldsymbol{ x}',j')\in \mathbb{ Z}_+^d\times S_0)$ is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as $\boldsymbol{ x}'$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.
Categories: math.PR