Abstract
We present a non-Gaussian local limit theorem for the number of occurrences of a given symbol in a word of length n generated at random. The stochastic model for the random generation is defined by a rational formal series with non-negative real coefficients. The result yields a local limit towards a uniform density function and holds under the assumption that the formal series defining the model is recognized by a weighted finite state automaton with two primitive components having equal dominant eigenvalue.
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Notes
- 1.
Here, for every interval \(I\subseteq \mathbb {R}\) and functions \(f,g :I \rightarrow \mathbb {C}\), by “\(g(t)=O(f(t))\) for \(t\in I\)” we mean “\(|g(t)| \le b |f(t)|\) for all \(t\in I\)”, for some constant \(b>0\).
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Goldwurm, M., Lin, J., Vignati, M. (2018). A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models. In: Konstantinidis, S., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2018. Lecture Notes in Computer Science(), vol 10952. Springer, Cham. https://doi.org/10.1007/978-3-319-94631-3_10
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DOI: https://doi.org/10.1007/978-3-319-94631-3_10
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