Information Theory is studied from the following view points: (1) the theory of entropy as amount of information; (2) the mathematical structure of information sources (probability measures); and (3) the theory of information channels. Shannon entropy and Kolmogorov-Sinai entropy are defined and their basic properties are examined, where the latter entropy is extended to be a linear functional on a certain set of measures. Ergodic and mixing properties of stationary sources are studied as well as AMS (asymptotically mean stationary) sources. The main purpose of this book is to present information channels in the environment of real and functional analysis as well as probability theory. Ergodic channels are characterized in various manners. Mixing and AMS channels are also considered in detail with some illustrations. A few other aspects of information channels including measurability, approximation and noncommutative extensions, are also discussed.
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"It is an interesting book that concentrates on a few themes, but treats them in depth and generality." -- Mathematical Reviews, 2003
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Inside This Book
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First Sentence:
In this chapter, basic ideas of entropy are presented from the works of Shannon and Kolmogorov-Sinai. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
ergodic channels, pointwise ergodic theorem, abstract measurable space, stationary channel, countable generator, ergodic sources, finite scheme, entropy functional, ergodic decomposition, statistical decision functions, probability measure space, measurable transformations, channel operators, relative entropy, measure preserving transformations, positive linear functional, coding theorems, monotone class, message spaces, stationary mean, algebraic models, compound state, mixing properties, finite partition, weak mixing Key Phrases - Capitalized Phrases (CAPs): (learn more)
Pointwise Ergodic Theorem, Bounded Convergence Theorem, Dominated Convergence Theorem, Mean Ergodic Theorem, Monotone Convergence Theorem, Jensen's Inequality, Kolmogorov-Sinai Theorem, Martingale Convergence Theorem, Riesz-Markov-Kakutani Theorem, S-invariant Baire New!
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Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
In this chapter, basic ideas of entropy are presented from the works of Shannon and Kolmogorov-Sinai. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
ergodic channels, pointwise ergodic theorem, abstract measurable space, stationary channel, countable generator, ergodic sources, finite scheme, entropy functional, ergodic decomposition, statistical decision functions, probability measure space, measurable transformations, channel operators, relative entropy, measure preserving transformations, positive linear functional, coding theorems, monotone class, message spaces, stationary mean, algebraic models, compound state, mixing properties, finite partition, weak mixing Key Phrases - Capitalized Phrases (CAPs): (learn more)
Pointwise Ergodic Theorem, Bounded Convergence Theorem, Dominated Convergence Theorem, Mean Ergodic Theorem, Monotone Convergence Theorem, Jensen's Inequality, Kolmogorov-Sinai Theorem, Martingale Convergence Theorem, Riesz-Markov-Kakutani Theorem, S-invariant Baire New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Front Flap | Table of Contents | First Pages | Index | Back Flap | Back Cover | Surprise Me!
Citations (learn more)
This book cites 30
books:
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books
cite this book:
- Real Analysis and Probability (Probability & Mathematical Statistics) by Robert B. Ash in Back Matter
- Foundations of Stochastic Analysis (Probability and Mathematical Statistics) by Malempati Madhusudana Rao in Back Matter
- Multidimensional Second Order Stochastic Processes (Series on Multivariate Analysis, Vol. 2) by Yuichiro Kakihara in Front Matter
- Maximum Entropy Models in Science and Engineering by J. N. Kapur in Back Matter
- The Theory of Bernoulli Shifts (Lectures in Mathematics) by Paul C. Shields in Back Matter
See all 30 books this book cites
- Stochastic Processes and Functional Analysis: A Volume of Recent Advances in Honor of M. M. Rao (Lecture Notes in Pure and Applied Mathematics) by Alan C. Krinik on page 207, and page 218
- Topics in Circular Statistics by S. Rao Jammalamadaka in Front Matter
- Mathematical Methods in Sample Surveys (Series on Multivariate Analysis, Vol 3) by Howard G. Tucker in Front Matter
- Structural Aspects in the Theory of Probability: A Primer In Probabilities On Algebraic-Topological Structures (Series on Multivariate Analysis, V. 7) by Herbert Heyer in Front Matter
- Linear Models: An Integrated Approach by Sreenivasa Rao Jammalamadaka in Front Matter
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