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.

Local limit laws for symbol statistics in bicomponent rational models / M. Goldwurm, J. Lin, M. Vignati. - (2021 Feb 18).

Local limit laws for symbol statistics in bicomponent rational models

M. Goldwurm;M. Vignati
2021

Abstract

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.
probability; formal languages and automata; limit distributions; local limit laws; pattern statistics; regular languages
Settore INF/01 - Informatica
Settore MAT/05 - Analisi Matematica
http://arxiv.org/abs/2102.09478v1
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/817456
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