Monotone additive statistics on heavy-tailed convolution semigroups

TL;DR

This paper characterizes monotone additive statistics on probability measure semigroups, showing that expectation is unique for measures with finite moments and that no non-zero statistics exist for the entire semigroup.

Contribution

It proves the uniqueness of expectation as the only monotone additive statistic on finite-moment measures and the non-existence of such statistics on the full semigroup.

Findings

Scalar multiples of expectation are the only monotone additive statistics with finite p-th moments.

No non-zero monotone additive statistics exist on the entire probability measure semigroup.

Abstract

We study sub-semigroups of the semigroup of probability measures on $\mathbb{R}$ and monotone additive statistics on them, by which we mean maps to the reals that are monotone with respect to the stochastic order and additive under convolution. We show that scalar multiples of the expectation are the unique monotone additive statistics on the semigroup of measures with finite $p$-th moment, for any $1 \le p < \infty$. We also prove that the entire semigroup of probability measures admits no non-zero monotone additive statistic at all.