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

Abstract: 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 ${\boldsymbol{Y}n}={(X{1,n},X_{2,n},J_n)}$ 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 $\boldsymbol{Y}, \boldsymbol{Y}'\in \mathbb{Z}+^2\times S_0$, 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 $\mathbb{Z}+^2\times S_0$ for the first time. For $\boldsymbol{Y}=(x_1,x_2,j)\in \mathbb{Z}+^2\times S_0$, 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 $x_1'$ and $x_2'$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.