Light-tailed asymptotics of GI/G/1-type Markov chains
Light-tailed asymptotics of GI/G/1-type Markov chains
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DOI：
10.3934/jimo.2017033
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发表时间：
2017-04
影响因子：
1.3
通讯作者：
Tatsuaki Kimura;H. Masuyama;Yutaka Takahashi
中科院分区：
文献类型：
--
作者：
Tatsuaki Kimura;H. Masuyama;Yutaka Takahashi
本文研究了GI/G/1型马尔可夫链平稳分布的轻尾渐近性。我们考虑三种情况：(ⅰ)尾部衰减率由与非边界层中的转移块矩阵\begin{document}$\{\boldsymbol{A}(k)相关联的某个参数\begin{document}$\theta$\end{document}决定；k=0,\pm1,\pm2,\dots\}$\end{document}； (二) 由边界层中过渡块矩阵 \begin{document}$\{\boldsymbol{B}(k);k=1,2,\dots\}$\end{document} 的生成函数的收敛半径； (ⅲ) 的收敛半径为 \begin{document}$\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$\end{document} 。在情况(一)中，我们将现有的M/G/1型马尔可夫链的渐近公式推广到GI/G/1型马尔可夫链。在情况（ⅱ）中，我们提出了一般的渐近公式，其中包括作为特殊情况的文献中的现有结果。在情况(ⅲ)中，我们推导出新的渐近公式。据我们所知，此类公式尚未见文献报道。
This paper studies the light-tailed asymptotics of the stationary distribution of the GI/G/1-type Markov chain. We consider three cases:(ⅰ) the tail decay rate is determined by a certain parameter \begin{document}$\theta$\end{document} associated with the transition block matrices \begin{document}$\{\boldsymbol{A}(k);k=0,\pm1,\pm2,\dots\}$\end{document} in the non-boundary levels; (ⅱ) by the convergence radius of the generating function of the transition block matrices \begin{document}$\{\boldsymbol{B}(k);k=1,2,\dots\}$\end{document} in the boundary level; and (ⅲ) by the convergence radius of \begin{document}$\sum_{k=1}^{\infty}z^k \boldsymbol{A}(k)$\end{document} . In the case (ⅰ), we extend the existing asymptotic formula for the M/G/1-type Markov chain to the GI/G/1-type one. In the case (ⅱ), we present general asymptotic formulas that include, as special cases, the existing results in the literature. In the case (ⅲ), we derive new asymptotic formulas. As far as we know, such formulas have not been reported in the literature.