On the Distribution of the Information Density of Gaussian Random Vectors: Explicit Formulas and Tight Approximations

Research output: Contribution to journalResearch articleContributedpeer-review



Based on the canonical correlation analysis, we derive series representations of the probability density function (PDF) and the cumulative distribution function (CDF) of the information density of arbitrary Gaussian random vectors as well as a general formula to calculate the central moments. Using the general results, we give closed-form expressions of the PDF and CDF and explicit formulas of the central moments for important special cases. Furthermore, we derive recurrence formulas and tight approximations of the general series representations, which allow efficient numerical cal-culations with an arbitrarily high accuracy as demonstrated with an implementation in PYTHON publicly available on GITLAB. Finally, we discuss the (in)validity of Gaussian approximations of the information density.


Original languageEnglish
Article number924
Number of pages29
Issue number7
Publication statusPublished - Jul 2022

External IDs

Scopus 85133729426
PubMed 35885147
Mendeley 8c63e9f1-7be9-3e4c-91a0-aeafff5ea215
unpaywall 10.3390/e24070924



  • Gaussian random vector, canonical correlation analysis, central moments, cumulative distribution function, information density, information spectrum, probability density function

Library keywords