Asymptotic property of the occupation measures in a multi-dimensional skip-free Markov modulated random walk

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