## 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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