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M. Abadi, S. Vaienti (2008)
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[Consider a stationary stochastic process X1, ..., X1. We study the distribution law of the number of occurrences of a string of symbols in this process. We consider three different ways to compute it. 1) When the string is the first block generated by the process, called Nt. 2) When the string is the first block generated by an independent, copy of the process, called Mt. 3) When the string \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{a} $$\end{document} is previously fixed, called Nt (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{a} $$\end{document}). We show how this laws can be approximated by a Poisson law with an explicit error for each approximation. We then derive approximations for the distribution of the first occurrence time of the string, according also to the above three different ways to chose it. The base of the proofs is the Chen-Stein method which is commonly used to prove approximations in total variation distance in the Markov context. We show how to apply it in non-Markovian mixing processes.]
Published: Jan 1, 2008
Keywords: Number of occurrences; occurrence time; Poisson approximation; exponential approximation; Chen-Stein method; mixing processes; 60F05; 60G10; 60G55; 37A50
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