We explore a unified $(k_1,k_2,\ldots,k_m)$-run in multi-state trials, examining its distributional properties and waiting time distribution. Our study reveals that this particular run serves as a generalization encompassing various patterns. Additionally, we discuss various results pertaining to existing patterns as special cases. To illustrate our findings, we provide an application related to DNA frequent patterns.

翻译：我们探讨了多状态试验中统一的$(k_1,k_2,\ldots,k_m)$-游程，研究了其分布特性与等待时间分布。我们的研究表明，该特定游程可作为涵盖多种模式的广义形式。此外，我们还将现有模式的相关结果作为特例进行了讨论。为阐明研究发现，我们提供了一个与DNA频繁模式相关的应用实例。