论文标题
在多维无跳跃马尔可夫调制随机行走中，职业措施的渐近性能
Asymptotic property of the occupation measures in a multi-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$ d $ -dimeNorial过程$ \ {\ boldsymbol {x} _n \} = \ {（x_ {1，n}，x_ {2，n}，...，...，x_ {d，n}，n}） $ s_0 $，其中各个进程$ \ {x_ {i，n} \}，i \ in \ {1,2，...，d \}，$是免费的。我们假设联合过程$ \ {\ boldsymbol {y} _n \} = \ {（\ boldsymbol {\ boldsymbol {x} _n，j_n）\} $是马克维亚人，而$ d of-d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ bolds的过渡概率$ \ {j_n \} $。假定该调制是均匀的空间。我们将此过程称为$ d $维无跳的马尔可夫调制随机步行。对于$ \ boldsymbol {y}，\ boldsymbol {y}'\ in \ mathbb {z} _+^d \ 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} $的预期访问次数，而该过程留下了非止痛面积$ \ mathbb {z} _+^_++^d d \ d \^d d \ d \ times s _ phise timess s_0 $。对于$ \ boldsymbol {y} =（\ boldsymbol {x}，j）\ in \ mathbb {z} _+^d \ times s_0 $，量度$（\ tilde {q} _ {\ boldsymbol {y} \ boldsymbol {y}'=（\ boldsymbol {x}'，j'）\ in \ mathbb {z} _+^d \ times s_0）$称为职业度量。我们的主要目的是将职业度量的渐近衰变率作为$ \ boldsymbol {x}'$沿给定方向送达无穷大。我们还获得了矩阵矩生成量度量度的收敛域。
We consider a discrete-time $d$-dimensional process $\{\boldsymbol{X}_n\}=\{(X_{1,n},X_{2,n},...,X_{d,n})\}$ 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 $\{\boldsymbol{Y}_n\}=\{(\boldsymbol{X}_n,J_n)\}$ is Markovian and that the transition probabilities of the $d$-dimensional process $\{\boldsymbol{X}_n\}$ 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 $\boldsymbol{y}, \boldsymbol{y}'\in \mathbb{Z}_+^d\times S_0$, 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 $\mathbb{Z}_+^d\times S_0$ for the first time. For $\boldsymbol{y}=(\boldsymbol{x},j)\in \mathbb{Z}_+^d\times S_0$, the measure $(\tilde{q}_{\boldsymbol{y},\boldsymbol{y}'}; \boldsymbol{y}'=(\boldsymbol{x}',j')\in \mathbb{Z}_+^d\times S_0)$ is called an occupation measure. Our primary aim is to obtain the asymptotic decay rate of the occupation measure as $\boldsymbol{x}'$ go to infinity in a given direction. We also obtain the convergence domain of the matrix moment generating function of the occupation measures.
