Exponential and Gaussian behavior in the tails of multivariate functions

TL;DR

This paper identifies that functions similar to a specific exponential form appear in the tails of many functions with properties like convexity and independence, linking these to Poisson processes and rare event conditioning.

Contribution

It demonstrates that the exponential-Gaussian tail behavior is a universal feature in a broad class of functions with certain geometric and invariance properties.

Findings

Functions in the tails resemble a specific exponential form

Connections to Poisson point processes near large sample edges

Insights into conditioning on rare events

Abstract

We observe that approximate copies of the function $\Lambda _{n}:\mathbb{R}^{n}\rightarrow (0,\infty )$ defined by \begin{equation*} \Lambda _{n}(x)=\exp \left( -x_{1}-\pi \sum_{i=2}^{n}x_{i}^{2}\right) \end{equation*} appear in the tails of a large class of functions, with properties related to coordinate independence, convexity, homotheticity, and homogeneity. The function $\Lambda _{n}$ is an entropy maximizer (on a half-space) that is uniquely determined by a homogeneity condition together with rotational invariance about the $x_{1}$ direction and its behavior near the origin. These results are connected to the limiting Poisson point processes found near the edges of large random samples, as well as the conditioning of random vectors on certain rare events, and can be thought of as variations of Laplace's method for estimating integrals.