AI 中文总结
研究吸收连续时间马尔可夫链转移函数的多项式衰减渐近性，给出系数矩阵秩为一的充分条件，并构造不可约反例说明不可约性不蕴含秩一渐近性。
AI 中文摘要
设$P(t)$为可数状态空间上吸收连续时间马尔可夫链的转移函数。我们研究形如$p_{ij}(t)\sim a_{ij} L(t)$（当$t\to\infty$时）的渐近关系，其中$L$与$i$和$j$无关。我们获得了系数矩阵$A=(a_{ij})$秩为一的一般条件，并描述了由此产生的生存概率和条件分布的后果。该分析适用于可约和不可约链。我们还构造了一个不可约反例，基于齐次树上的被杀随机游走，其中存在共同的渐近尺度但系数矩阵的秩大于一。这表明不可约性本身并不蕴含秩一渐近性。这些结果确定了固定状态转移渐近性决定整个转移函数渐近行为的条件，并阐明了在缺乏额外结构时此类结论的局限性。
英文摘要
Let $P(t)$ be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form $p_{ij}(t)\sim a_{ij} L(t)$, $t\to\infty$, where $L$ is independent of $i$ and $j$. We obtain general conditions under which the coefficient matrix $A=(a_{ij})$ has rank one and describe the resulting consequences for survival probabilities and conditional distributions. The analysis applies to both reducible and irreducible chains. We also construct an irreducible counterexample, based on a killed random walk on a homogeneous tree, for which a common asymptotic scale exists but the coefficient matrix has rank greater than one. This shows that irreducibility alone does not imply rank-one asymptotics. The results identify conditions under which fixed-state transition asymptotics determine the asymptotic behaviour of the entire transition function and clarify the limitations of such conclusions in the absence of additional structure.
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