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The entropy of functions of finite-state Markov chains. (English) Zbl 0085.12401

Trans. 1st Prague Conf. Information theory, statistical decision functions, random processes, Liblice, 1956, 13-20 (1957).
This paper is best dealt with by quoting the author’s own summary: It is shown that the entropy \(H\) of an ergodic stationary process \(\{y_n, -\infty < n < \infty\}\) with states \(a = 1, \ldots, A\) such that \(y_n = \Phi(x_n)\), where \(\{x_n\}\) is a stationary ergodic finite-state Markov process with states \(i=1, \ldots, I\) and transition matrix \(M =\|(m(i, j)\|\) is given by \[ H = -\int\sum r_a(w) \log r_a(w)\, dQ(w), \] where \(r_a\) is a function, defined on the set \(W\) of all \(w = (w_1,\ldots, w_I)\) such that \(w_i\geq 0\), \(\sum_{i=1}^I w_i=1\), by \(r_a(w)=\sum_{i,j\in\Phi}\sum_{(j)=a}(j)w_im(i,j)\) and \(Q\) is the conditional distribution of \(x_0\) given \(y_0, y_{-1}, \ldots\). An integral equation is obtained for \(Q\), and a method is given for showing, under rather strong hypotheses, that the solution of this integral equation is unique. An example in which \(Q\) is singular is given.
Reviewer: S. Vajda

MSC:

60J25 Continuous-time Markov processes on general state spaces
60G30 Continuity and singularity of induced measures

Citations:

Zbl 0085.12305