We consider a discrete-time d-dimensional process \(\{{\varvec{X}}_n\}=\{(X_{1,n},X_{2,n},\ldots ,X_{d,n})\}\) on \({\mathbb {Z}}^d\) with a background process \(\{J_n\}\) on a countable set \(S_0\), where individual processes \(\{X_{i,n}\},i\in \{1,2,\ldots ,d\},\) are skip free. We assume that the joint process \(\{{\varvec{Y}}_n\}=\{({\varvec{X}}_n,J_n)\}\) is Markovian and that the transition probabilities of the d-dimensional process \(\{{\varvec{X}}_n\}\) 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-modulated random walk. For \({\varvec{y}}, {\varvec{y}}'\in {\mathbb {Z}}_+^d\times S_0\), consider the process \(\{{\varvec{Y}}_n\}_{n\ge 0}\) starting from the state \({\varvec{y}}\) and let \({\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}\) be the expected number of visits to the state \({\varvec{y}}'\) before the process leaves the nonnegative area \({\mathbb {Z}}_+^d\times S_0\) for the first time. For \({\varvec{y}}=({\varvec{x}},j)\in {\mathbb {Z}}_+^d\times S_0\), the measure \(({\tilde{q}}_{{\varvec{y}},{\varvec{y}}'}; {\varvec{y}}'=({\varvec{x}}',j')\in {\mathbb {Z}}_+^d\times S_0)\) is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as \({\varvec{x}}'\) goes to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measure.

我们考虑离散时间d维过程\（\ {{\ varvec {X}} _ n \} = \ {（X_ {1，n}，X_ {2，n}，\ ldots，X_ {d，n }）\} \）在\（{\ mathbb {Z}} ^ d \）上，而背景进程\（\ {J_n \} \）在可数集合\（S_0 \）上，其中单个进程\（\ { \ {1,2，\ ldots，d \}，\）中的X_ {i，n} \}，i \是免费跳过的。我们假设联合过程\（\ {{\ varvec {Y}} _ n \} = \ {（{\ varvec {X}} _ n，J_n）\} \）是马尔可夫式，并且d的转移概率-尺寸过程\（\ {{\ varvec {X}} _ n \} \）根据后台过程\（\ {J_n \} \）的状态而变化。假定该调制是空间均匀的。我们将此过程称为d维无跳跃马尔可夫调制随机游动。对于\（{\ varvec {y}}，{\ varvec {y}}'\ {{mathbb {Z}} _ + ^ d \ times S_0 \）中的\（\ {{\ varvec {Y }} _ n \} _ {n \ ge 0} \）从状态\（{\ varvec {y}} \）开始并让\（{\ tilde {q}} _ {{\ varvec {y}}， {\ varvec {y}}'} \）是进程离开非负区域\（{\ mathbb {Z}} _ +之前对状态\（{\ varvec {y}}'\）的预期访问次数^ d \次S_0 \）。对于\（{\ varvec {y}} =（{{varvec {x}}，j）\ {{mathbb {Z}} _ + ^ d \ times_0_0）中的度量\（（{{tilde {q}} _ {{\ varvec {y}}'}; {\ varvec {y}}'}; {\ varvec {y}}'=（{\ varvec {x}}'，j '）\在{\ mathbb {Z}} _ + ^ d \ times S_0）\）中被称为占领措施。我们的主要目的是获得占领度量的渐近衰减率，因为\（{\ varvec {x}}'\）在给定方向上达到无穷大。我们还获得了占用度量的矩阵矩生成函数的收敛域。