The Cauchy Distribution in Information Theory.

Area of Science:

• Information Theory
• Probability Theory
• Statistical Mechanics

Background:

• The Gaussian law is foundational in information theory for analog random variables.
• Existing information theoretic results are predominantly based on Gaussian distributions.

Purpose of the Study:

• To explore information theoretic results for Cauchy distributions.
• To introduce and analyze new concepts relevant to Cauchy distributions.

Main Methods:

• Theoretical analysis of probability measures.
• Introduction of novel concepts: equivalent pairs of probability measures and strength of real-valued random variables.

Main Results:

• Demonstration of information theoretic results analogous to Gaussian counterparts for Cauchy distributions.
• Establishment of the relevance of equivalent pairs and variable strength to Cauchy distributions.

Conclusions:

• Cauchy distributions possess rich information theoretic properties.
• The newly introduced concepts provide valuable tools for analyzing Cauchy distributions.