We consider a (multidimensional) additive functional of semi-Markov chain, defined by an ergodic Markov chain with a finite number of states. The distribution of random vectors, governing the process, is supposed to be lattice and light-tailed. We derive the exact asymptotics in the local limit theorem. As a consequence, we establish a local central limit theorem.

中文翻译：

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半马尔可夫链的（多维）加法泛函的局部定理

我们考虑半马尔可夫链的（多维）可加泛函，由具有有限状态数的遍历马尔可夫链定义。控制该过程的随机矢量分布应该是晶格且轻尾的。我们在局部极限定理中得出精确的渐近性。结果，我们建立了局部中心极限定理。