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Summary
This summary is machine-generated.

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.

Keywords:
Rényi entropyestimation errorsfractional momentsguessingintegral representationjamminglogarithmic expectationmoment-generating functionmultivariate Cauchy distributions

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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.