A note on generalizations of Shannon-McMillan theorem (Q775646)

This paper is a continuation of an earlier paper of the author [Pac. J. Math. 11, 705--714 (1961; Zbl 0102.13101)] and makes use of the same notations. As before, there are considered two probability measures \(\nu\) and \(\mu\) on the infinite product \(\sigma\)-algebra of subsets of the infinite product space \(\Omega = \pi X\), where \(\nu\) is assumed to be stationary and \(\mu\) to be Markovian with stationary transition probabilities. For the contractions \(\nu_{m,n}\) and \(\mu_{m,n}\) of \(\nu\) and \(\mu\) to \(F_{m,n}\) (\(\sigma\)-algebra corresponding to the ``coordinates'' of \(m\) to \(n\)) it is assumed that \(\nu_{m,n} \ll \mu_{m,n}\), \(f_{m,n}\) being the corresponding Radon-Nikodym derivative. In the place of the condition \[ \int(\log f_{0,n} - \log f_{0,n-1}) \,d\nu \le M \quad\text{for }n= 1, 2, \ldots, \tag{1}\] \((M\) finite number), which together with \(\int(\log f_{0,0}\,d\nu < \infty\) implies the \(L_1(\nu)\) convergence of \(\{\pi^{-1} \log f_{0,n}\}\), is now used the stronger condition \[ \int \frac{f_{0,n}}{f_{0,n-1}} \,d\nu \le L, \quad\text{for }n= 1, 2, \ldots, \tag{2}\] for the proof of the convergence with \(\nu\)-probability one of the same sequence. Applying a theorem of \textit{L. Breiman} [Ann. Math. Stat. 28, 809--811 (1957; Zbl 0078.31801)] the author derives the latter convergence from the condition \[ \int \sup_{k\ge 1} \log\frac{f_{-k,0}}{f_{-k,-1}}\,d\nu < \infty \tag{3} \] which is implied by condition (2), according to Theorem 1. In the case the ``alphabet'' \(X\) is countable and the one-dimensional ordinary entropy \(H_1\) corresponding to \(\nu\) is finite, from the above results it is possible to deduce (by a suitable choice of the dominating measure \(\mu\)) the \(L_1(\nu)\) convergence [\textit{L. Carleson}, Math. Scand. 6, 175--180 (1958; Zbl 0088.10603)] and the convergence with \(\nu\)-probability one [\textit{K. L. Chung}, Ann. Math. Stat. 32, 612--614 (1961; Zbl 0115.35503)] of the sequence \(\{\pi^{-1} \log P(x_0, x_1, \ldots, x_n)\}\), where \(P(x_0, x_1, \ldots, x_n)\) gives the probability corresponding to \(\nu\) of blocks of \( n+1\) letters from \(X\). Moreover, by using a similar approach, the author gives a sharpened version of Carleson's and Chung's theorems above, where it is possible to replace the condition of finiteness of \(H_1\) by a weaker condition, which consists in assuming that the (ordinary) entropy rate corresponding to the stationary measure \(\nu\) is finite.