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.
Categories: math.PR