A Short Course on the Geometry of Probability Distributions

In the late 19th Century, Felix Klein, in his Erlangen Program, proposed the view that geometry was the collection of properties of a space which are invariant under a given transformation group. Elie Cartan’s work later allowed the expansion of this notion from spaces to more general mathematical objects.

In the case of univariate probability distributions, the geometry we refer to is the set of properties preserved by the “location scale” group–the affine group on the line.

The differential invariants which describe this geometry can be calculated with elementary methods. Their properties reveal previously unknown results including new limit theorems and new insights into Extreme Value Theory. In particular, this approach identifies “exceptional” distributions–those whose differential invariants take constant values.

The topics for the course include: The solution of the equivalence problem for univariate distributions preserved by the action of the affine group on the line. Omega Functions, a new approach to distributions with finite means and their associated statistics. Deformations of Omega Functions, a new limit theorem and a collection of distributions defined by rational functions which interpolate between the Bernoulli distribution and the limit theorem’s attractor. A new characterisation of the Extreme Value Distributions which is easily implemented and includes an intrinsic method of measuring the degree of convergence to an Extreme Value attractor. A new duality between Extreme Value Distributions and Generalised Pareto distributions.

The course should be of interest to students of Probability for its novel distributions and new insights into Extreme Value theory. For students of Geometry, the course presents a very general sort of Equivalence Problem that can nevertheless be solved by elementary calculations and illustrates the power of Cartan’s approach to differential invariants.

Outline of Lectures
1. Course overview and the equivalence problem for the affine group on the line (location scale transformations).
2. Omega functions, definition, examples, properties and associated statistics.
3. Deformations of the Omega function for the Bernoulli distribution and a new Central Limit Theorem.
4. A new synthesis of Extreme Value Theory results via differential invariants. Duality between Extreme Value attractors and Generalised Pareto distributions.
5. Proof of a new Extreme Value Theorem.
15 May 2023