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 $\{{\mathbf{Y}}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$ 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 $\mathbf{Y},{\mathbf{Y}}^{\prime }\in {\mathbb{Z}}_{+}^2\times S_0$, consider the process $\{{\mathbf{Y}}_n{\}}_{n\ge 0}$ starting from the state $\mathbf{Y}$ and let ${\tilde{q}}_{\mathbf{Y},{\mathbf{Y}}^{\prime }}$ be the expected number of visits to the state ${\mathbf{Y}}^{\prime }$ before the process leaves the nonnegative area ${\mathbb{Z}}_{+}^2\times S_0$ for the first time. For $\mathbf{Y}=(x_1,x_2,j)\in {\mathbb{Z}}_{+}^2\times S_0$, the measure $({\tilde{q}}_{\mathbf{Y},{\mathbf{Y}}^{\prime }};{\mathbf{Y}}^{\prime }=(x_1^{\prime },x_2^{\prime },j^{\prime })\in {\mathbb{Z}}_{+}^2\times S_0)$ is called an occupation measure. Our main aim is to obtain asymptotic decay rate of the occupation measure as the values of $x_1^{\prime }$ and $x_2^{\prime }$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.

23 pages, 2 figures