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Monotone additive statistics on heavy-tailed convolution semigroups

Tobias Fritz, Xiaosheng Mu, Omer Tamuz

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.

Monotone additive statistics on heavy-tailed convolution semigroups

Abstract

We study sub-semigroups of the semigroup of probability measures on 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 -th moment, for any . We also prove that the entire semigroup of probability measures admits no non-zero monotone additive statistic at all.

Paper Structure

This paper contains 11 sections, 8 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Related literature
  3. Main definitions and results
  4. Proof of Theorem \ref{['thm:L1']}: Uniqueness of the expectation
  5. Proof of Theorem \ref{['thm:PR']}
  6. Proof of Theorem \ref{['thm:Lpsi']}
  7. Other intermediate semigroups
  8. Extensions of the expectation
  9. Further examples
  10. Overview of semigroups considered
  11. Open questions

Key Result

Theorem 1

On the following sub-semigroups of $\mathcal{P}$, the monotone additive statistics $\varphi$ are precisely the maps of the form $\varphi(\mu) = c { \mathbb{E}\left[{\mu}\right] }$ for some $c \geq 0$:

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 5 more