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

We consider a discrete-time two-dimensional process $\{(X_{1,n},X_{2,n})\}$ on ${\mathbb{Z}}^2$ with a background process $\{J_n\}$ on a finite set $S_0$, where individual processes $\{X_{1,n}\}$ and $\{X_{2,n}\}$ are both skip free. We assume that the joint process $\{{\mathit{Y}}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(X_{1,n},X_{2,n})\}$ 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 two-dimensional skip-free Markov modulate random walk. For $\boldsymbol{Y}, \boldsymbol{Y}‘\in $\mathbb{Z}$+^2\times S_0$,considertheprocess${\boldsymbol{Y}n}{n\ge 0}$startingfromthestate$\boldsymbol{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 ${\mathbb{Z}}_{+}^2\times S_0$ for the first time. For $\mathit{Y}=(x_1,x_2,j)\in {\mathbb{Z}}_{+}^2\times S_0$, the measure $(\tilde{q}{\boldsymbol{Y},\boldsymbol{Y}’}; \boldsymbol{Y}‘=(x_1’,x_2’,j’)\in $\mathbb{Z}$_+^2\times S_0)$ is called an occupation measure. Our main aim is to obtain asymptotic decay rate of the occupation measure as the values of $x_1^{\mathrm{\prime }}$ and $x_2^{\mathrm{\prime }}$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.