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Some informational properties of Markov purejump processes. (English) Zbl 0666.60076

Let \(X_ t\), \(t\geq 0\), be a continuous-time ergodic irreducible Markov chain with a finite state space, \(Q=(q_{ij})\) the density matrix. By means of the embedded chain, the limit average entropy is calculated to be \[ \rho^{-1}\sum Q^{ij}q_{ij}(1-\ln q_{ij}), \] where \(Q^{ij}\) is the (i,j) cofactor of Q and \(\rho\) is the product of the non-zero eigenvalues of Q. Taking the imbedded chain as the information source, the author shows the Shannon-McMillan’s theorem, i.e. the equipartition property, is valid.
Besides, the limit average cross-entropy between two Markov chains with density matrices \(Q=(q_{ij})\) and \(R=(r_{ij})\), respectively, is also calculated to be \[ \rho_ Q^{-1}[\sum_{i\neq j}Q^{ij}q_{ij}\ln (q_{ij}/r_{ij})-\sum^{s}_{i=1}Q^{ii}(q_ i-r_ i)]. \]
Reviewer: He Shengwu

MSC:

60J75 Jump processes (MSC2010)
94A17 Measures of information, entropy
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