二维无跳马尔可夫调制随机游动中占领测度的渐近性质
Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk
摘要：
我们考虑一个离散时间二维过程$\{(X_{1,n},X_{2,n})\}$在$\mathbb{Z}^2$上，背景过程$\{J_n\}$在有限集合$S_0$上，其中单个过程$\{X_{1,n}\}$和$\{X_{2,n}\}$都是无跳的。我们假设联合过程$\{\boldsymbol{Y}_n\}=\{(X_{1,n},X_{2,n},J_n)\}$是马尔可夫的，并且二维过程$\{(X_{1,n},X_{2,n})\}$的转移概率根据背景过程$\{J_n\}$的状态而变化。这种调制被假定为空间均匀的。我们把这个过程称为二维无跳马尔可夫调制随机游动。对于$\boldsymbol{Y}, \boldsymbol{Y}'\in \mathbb{Z}_+^2\times S_0$，考虑从状态$\boldsymbol{Y}$开始的过程$\{\boldsymbol{Y}_n\}_{n\ge 0}$，并设$\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}$为过程第一次离开非负区域$\mathbb{Z}_+^2\times S_0$之前访问状态$\boldsymbol{Y}'$的预期次数。对于$\boldsymbol{Y}=(x_1,x_2,j)\in \mathbb{Z}_+^2\times S_0$，度量$(\tilde{q}_{\boldsymbol{Y},\boldsymbol{Y}'}; \boldsymbol{Y}'=(x_1',x_2',j')\in \mathbb{Z}_+^2\times S_0)$称为占用度量。我们的主要目的是获得渐近衰减率的占领措施的$x_1'$和$x_2'$的值在一个给定的方向走向无穷大。我们还得到了占有测度的矩阵矩母函数的收敛域。
英文摘要：
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.
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