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Some Useful Integral Representations for Information-Theoretic Analyses
1The Andrew and Erna Viterbi Faculty of Electrical Engineering, Israel Institute of Technology Technion City, Haifa 3200003, Israel.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
This study introduces a new integral representation for fractional moments of random variables. This method simplifies calculations, especially for sums of many variables, by reducing multi-dimensional integration.
Area of Science:
- Probability Theory
- Information Theory
- Mathematical Analysis
Background:
- Extends prior work on integral representations for logarithmic functions.
- Addresses the need for efficient calculation of fractional moments.
Purpose of the Study:
- Derive an exact integral representation for fractional moments.
- Apply this representation to information-theoretic problems.
- Demonstrate computational advantages over traditional methods.
Main Methods:
- Developed a novel one- or two-dimensional integral representation.
- Applied the formula to various examples, including sums of independent random variables.
- Focused on moments of non-integer order (fractional moments).
Main Results:
- Obtained compact, exact formulas for fractional moments.
- Showcased simplified numerical evaluations for information-theoretic quantities.
- Demonstrated significant computational efficiency for large sums of random variables.
Conclusions:
- The proposed integral representation offers a powerful tool for calculating fractional moments.
- This method substantially reduces computational complexity compared to direct integration.
- Facilitates easier numerical evaluation in practical applications.
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