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

Abstract

We consider a discrete-time $d$-dimensional process $\{{\mathbf{X}}_n\}=\{(X_{1,n},X_{2,n},...,X_{d,n})\}$ on ${\mathbb{Z}}^d$ with a background process $\{J_n\}$ on a countable set $S_0$, where individual processes $\{X_{i,n}\},i\in \{1,2,...,d\},$ are skip free. We assume that the joint process $\{{\mathbf{Y}}_n\}=\{({\mathbf{X}}_n,J_n)\}$ is Markovian and that the transition probabilities of the $d$-dimensional process $\{{\mathbf{X}}_n\}$ 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 $\mathbf{y},{\mathbf{y}}^{\prime }\in {\mathbb{Z}}_{+}^d\times S_0$, 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 ${\mathbb{Z}}_{+}^d\times S_0$ for the first time. For $\mathbf{y}=(\mathbf{x},j)\in {\mathbb{Z}}_{+}^d\times S_0$, the measure $({\tilde{q}}_{\mathbf{y},{\mathbf{y}}^{\prime }};{\mathbf{y}}^{\prime }=({\mathbf{x}}^{\prime },j^{\prime })\in {\mathbb{Z}}_{+}^d\times S_0)$ is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as ${\mathbf{x}}^{\prime }$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.

33 pages, 3 figures. This manuscript partially overlaps with arXiv:1611.02434 and arXiv:2001.00700