We study the local limit distribution of the number of occurrences of a symbol in words of length $n$ generated at random in a regular language according to a rational stochastic model. We present an analysis of the main local limits when the finite state automaton defining the stochastic model consists of two primitive components. The limit distributions depend on several parameters and conditions, such as the main constants of mean value and variance of our statistics associated with the two components, and the existence of communications from the first to the second component. The convergence rate of these results is always of order $O(n^{-1/2})$. We also prove an analogous $O(n^{-1/2})$ convergence rate to a Gaussian density of the same statistic whenever the stochastic models only consists of one (primitive) component.

中文翻译：

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双组分有理模型中符号统计的局部极限定律

我们根据合理的随机模型研究了在常规语言中随机生成的长度为$ n $的单词中符号出现次数的局部极限分布。当定义随机模型的有限状态自动机由两个基本成分组成时，我们对主要局部极限进行了分析。极限分布取决于几个参数和条件，例如平均值的主要常数和与这两个组件相关的统计量的方差，以及从第一组件到第二组件的通信的存在。这些结果的收敛速度始终为$ O（n ^ {-1/2}）$。每当随机模型仅包含一个（原始）分量时，我们还证明了相似的$ O（n ^ {-1/2}）$收敛速度到相同统计量的高斯密度。