A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models

Massimiliano Goldwurm,Jianyi Lin,Marco Vignati

doi:10.1007/978-3-319-94631-3_10

Massimiliano Goldwurm, Jianyi Lin + Show 1 more

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https://doi.org/10.1007/978-3-319-94631-3_10

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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.

Highlights

A Local Limit Property for Pattern Statistics in Bicomponent Stochastic ModelsTo cite this version: Massimiliano Goldwurm, Jianyi Lin, Marco Vignati
Denoting by Yn such a random variable, traditional goals of interest are the asymptotic evaluation of mean value and variance of Yn as well as its limit distribution and local limit evaluations for its probability function
We consider the rational stochastic models, which are defined by rational formal series in non-commutative variables with coefficients in R+ [3]

Summary

A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models

To cite this version: Massimiliano Goldwurm, Jianyi Lin, Marco Vignati. A Local Limit Property for Pattern Statistics in Bicomponent Stochastic Models. 20th International Conference on Descriptional Complexity of Formal Systems (DCFS), Jul 2018, Halifax, NS, Canada. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. (1) Dipartimento di Matematica, Universita degli Studi di Milano, Milano – Italy (2) Department of Mathematics, Khalifa University, Abu Dhabi - United Arab Emirates

Introduction

Preliminary notions

Primitive case

Bicomponent models

Analysis of the characteristic function

Main result

Conclusions