Generalizations of Shannon-McMillan theorem
type: Item
instance of: scholarly article
title: Generalizations of Shannon-McMillan theorem
zbMATH Open document ID: 0102.13101
author: Shu-Teh C. Moy
published in: Pacific Journal of Mathematics
publication date: 1961-01-01T00:00:00Z
review text: This paper represents an extension of the work of the present reviewer concerning different generalizations of Shannon-McMillan's limit theorem in the case of abstract alphabets. [The exact name of the second, No. 6, between the three related references cited by the author, is: ``Notions généralisées d'incertitude, d'entropie et d'information du point de vue de la théorie de martingales'' where the numbers of pp. are interchanged with those of the reference No. 5]. Let \((X, S)\) be a measurable space and \((\Omega = \prod_{n=-\infty}^\infty X_n\), \(F = \prod_{n=-\infty}^\infty S_n\)) be the corresponding infinite product of this space. If \(\omega = \{\ldots, x_{-1}, x_0, x_1, \ldots\}\) is an element of the space \(\Omega\), then \(x_n\) is called the \(n\)-th coordinate of \(\omega\) and shall be considered as a function of \(\Omega\) to \(X\). Let \(\mu\) and \(\nu\) be two probability measures defined on the infinite product \(\sigma\)-algebra \(F\). Let \((\Omega_n = \prod_{i=1}^n X_i\), \(F_n = \prod_{i=1}^n S_i\)) be the \(n\)-product measurable space of \((X,S)\), corresponding to the coordinates of order 1 to \(n\), and let \(F_{m,n}\), \(m\le n\), \(n = 0, \pm 1, \pm 2, \ldots\) be the \(\sigma\)-algebra of subsets of \(\Omega\) consisting of sets of the form \[ [\omega = \{\ldots, x_{-1}, x_0, x_1, \ldots\} : (x_m, x_{m+1}, \ldots, x_n) \in E] \] where \(E\in F_{n-m+1}\). Let \(F_{-\infty,n}\) be the \(\sigma\)-algebra generated by \(\cup_{m=-1}^{-\infty} F_{m,n}\). Let \(\mu_{m,n}\), \(\nu_{m,n}\) be the contractions of \(\mu\), \(\nu\), respectively, to \(F_{-\infty,n}\) and \(\mu_{-\infty,n}\), \(\nu_{-\infty,n}\)be the con-tractions of \(\mu\), \(\nu\), respectively, to \(F_{-\infty,n}\). Throughout this paper \(\nu_{m,n}\) is assumed to be absolutely continuous with respect to \(\mu_{m,n}\), \(\nu_{m,n}\ll \mu_{m,n}\) for \(m < n\), \(n=0, \pm 1, \pm 2, \ldots\). Let \(f_{m,n}\) be the derivative of \(\nu_{m,n}\) with respect to \(\mu_{m,n}\) so th…
Mathematics Subject Classification ID: 94A24
zbMATH DE Number: 3166609
MaRDI profile type: Publication
full work available at URL: pjm.1961.11.705
OpenAlex ID: W2043323813
DOI: 10.2140/PJM.1961.11.705
description: scientific article; zbMATH DE number 3166609