Large deviations for heavy-tailed random elements in convex cones

TL;DR

This paper establishes large deviation principles for sums of heavy-tailed random elements within various convex cones, using a novel approach that avoids embedding cones into linear spaces, applicable to diverse mathematical structures.

Contribution

It introduces a new technique for large deviations in convex cones that does not rely on linear embedding, extending results to broader classes of cones.

Findings

Large deviation results for heavy-tailed elements in convex cones.

Technique applicable to cones like convex sets, half-line with maximum, and function spaces.

Avoids embedding cones into linear spaces, broadening applicability.

Abstract

We prove large deviation results for sums of heavy-tailed random elements in rather general convex cones being semigroups equipped with a rescaling operation by positive real numbers. In difference to previous results for the cone of convex sets, our technique does not use the embedding of cones in linear spaces. Examples include the cone of convex sets with the Minkowski addition, positive half-line with maximum operation and the family of square integrable functions with arithmetic addition and argument rescaling.