论文标题
二维无跳跃马尔可夫的渐近特性
Asymptotic property of the occupation measures in a two-dimensional skip-free Markov modulated random walk
论文作者
论文摘要
我们考虑一个离散的时间二维过程$ \ {（x_ {1，n}，x_ {2，n}）$ \} $ on $ \ mathbb {z}^2 $，带有后台过程$ \ {j_n \ \ { $ \ {x_ {2，n} \} $都是免费的。我们假设关节过程$ \ {\ boldsymbol {y} _n \} = \ {（x_ {1，n}，x_ {2，n}，n}，j_n）\} $是马克维亚人，是二维过程的过渡概率，二维过程$ \ \ \ \ \ {（x_______）到背景过程的状态$ \ {j_n \} $。假定该调制是均匀的空间。我们将此过程称为二维无跳跃马尔可夫调制随机步行。对于$ \ boldsymbol {y}，\ boldsymbol {y}'\ in \ mathbb {z} _+^2 \ times s_0 $，请考虑过程$ \ {\ boldsymbol {y} _n} _n \} _n \} _ {n \ ge 0} $从状态开始$ \ tilde {q} _ {\ boldsymbol {y}，\ boldsymbol {y}'} $是在此过程之前对状态$ \ boldsymbol {y} $的预期访问次数，而该过程留下了非止痛面积$ \ naterbb $ \ mathbb {z} _++^_+^_+^2 \^2 \ times s_____。对于$ \ boldsymbol {y} =（x_1，x_2，j）\ in \ mathbb {z} _+^2 \ times s_0 $，量度$（\ tilde {q} _ {\ boldsymbol \ boldsymbol {y}'=（x_1'，x_2'，j'）\ in \ mathbb {z} _+^2 \ times s_0）$称为职业度量。我们的主要目的是获得职业量度的渐近衰减率，作为$ x_1'$和$ x_2' $的值，以给定的方向向无穷大。我们还获得了矩阵矩生成量度量度的收敛域。
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.
