Generalizations of Shannon-McMillan theorem
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 that \(f_{m,n}\) is \(F_{m,n}\) measurable and positive with \(\nu\)-probability one. The main theorem proved by the author is the following (Theorem 5): If \(\nu\) is stationary and \(\mu\) is Markovian with stationary transition probabilities and if \(\int \log f_{0,0}\, d\nu < \infty\) and if there is a finite number \(M\) such that if \(\int (\log f_{0,n} - \log f_{0,n-1})\, d\nu \le M\) for \(n = 1, 2, \ldots\) then \(n^{-1}\log f_{0,n}\) converges in \(L_1(\nu)\) as \(n\to\infty\). In particular, if \(\nu\) is ergodic, the limit is equal to a nonnegative constant with \(\nu\)-probability one. This result represents an extension of our respective result because in our case the dominating measure \(\mu\) above was assumed to be stationary of the product (independent) type. As in our case, the crucial step in establishing this result is to prove the \(L_1(\nu)\) convergence of \(\{\log f_{-n,0} - \log f_{-n,-1}\}\) and this is made in a similar manner by using a property representing an extended version of our ``\(A_\mu\) condition'' \((\nu_{-\infty,m} \ll \nu_{-\infty,0}\times \mu_{1,n}\), \(n =1, 2, \ldots)\), which, as the author observes, is a consequence of the other assumptions, so that it is not necessary to be formulated as a condition. The example given by the author at the end of his paper is not in contradiction with our statement that, in our case, the condition \(A_\mu\) is necessary for the \(L_1(\nu)\) convergence of \(\{\log f_{-n,0} - \log f_{-n,-1}\}\) because in this example our assumption is not respected that the dominating measure \(\mu\) is of the product (independent) type.
scientific article; zbMATH DE number 3166609