Generalizations of Shannon-McMillan theorem
نویسندگان
چکیده
منابع مشابه
A Bilateral Version of the Shannon-McMillan-Breiman Theorem
We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by CT n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tmα ∨ T−1α ∨ . . . ∨ T−nα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ ( −1 n+m ) logμ(C n,m,α(x)) = hμ(T, α), where μ is a T ...
متن کاملA Bilateral version of Shannon-Breiman-McMillan Theorem
We give a new version of the Shannon-McMillan-Breiman theorem in the case of a bijective action. For a finite partition α of a compact set X and a measurable action T on X, we denote by C n,m,α(x) the element of the partition α ∨ T 1α ∨ . . . ∨ Tα ∨ T−1α ∨ . . . ∨ Tα which contains a point x. We prove that for μ-almost all x, lim n+m→∞ (
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We formulate and prove a quantum Shannon-McMillan theorem. The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on Z-lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes. The one-dimensional case covers quantum information sources and is basic for coding theorems.
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By means of the notion of likelihood ratio, the limit properties of the sequences of arbitrarydependent continuous random variables are studied, and a kind of strong limit theorems represented by inequalities with random bounds for functions of continuous random variables is established. The Shannon-McMillan theorem is extended to the case of arbitrary continuous information sources. In the pro...
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ژورنال
عنوان ژورنال: Pacific Journal of Mathematics
سال: 1961
ISSN: 0030-8730,0030-8730
DOI: 10.2140/pjm.1961.11.705