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

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. Using the series representations we give closed-form expressions of the PDF and CDF for important special cases and derive tight approximations for the general case. Furthermore, we derive recurrence formulas, which allow very efficient numerical calculations 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.