Exponential-Uniform Identities Related to Records

TL;DR

This paper explores a geometric structure derived from planar Poisson processes, revealing symmetry properties that lead to new distributional identities involving exponential and uniform variables.

Contribution

It introduces a novel geometric framework and uncovers distributional identities connecting exponential and uniform random variables through grid area symmetries.

Findings

Identifies symmetry properties of grid areas from Poisson processes.

Derives distributional identities for rational functions of exponential and uniform variables.

Establishes connections between geometric structures and probabilistic identities.

Abstract

We consider a rectangular grid induced by the south-west records from the planar Poisson point process in $R^2_+$. A random symmetry property of the matrix whose entries are the areas of tiles of the grid implies cute multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.