The medians for exponential families and the normal law

TL;DR

This paper characterizes the standard Gaussian distribution as the unique exponential family where the parameter t is a median of the distribution P_t, using the Choquet Deny equation.

Contribution

It provides a novel characterization of the Gaussian law within exponential families based on median properties and the Choquet Deny equation.

Findings

Only the Gaussian law has median t for all P_t in the exponential family.

The proof employs the Choquet Deny equation to establish the characterization.

The result uniquely identifies the standard Gaussian distribution among exponential families.

Abstract

Let $P$ a probability on the real line generating a natural exponential family $(P_t)_{t\in \R}$. We show that $t$ is a median of $P_t$ for all $t$ only if $P$ is the standard Gaussian law $N(0.1).$ The proof is based on the Choquet Deny equation.