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On convex order and supermodular order without finite mean

Benjamin Côté, Ruodu Wang

TL;DR

This paper extends the classical convex order and its relation to the supermodular order to random variables without finite means. It clarifies that two widely used definitions of convex order, cx and cx^dagger, coincide on finite-mean spaces but diverge when infinite means are allowed, with antisymmetry failing for cx^dagger in general. The authors prove that the fundamental two-variable bound involving counter-monotone and comonotone sums, X^{ct}+Y^{ct} ≤_cx X+Y ≤_cx X^{co}+Y^{co}, remains valid in the infinite-mean setting and extend analogous results to arbitrary dimensions. They also connect these results to optimal transport, provide ES-based and quantile-characterization conditions for infinite-mean convex order, and discuss multivariate nuances of the supermodular and concordance orders, highlighting cases where standard finite-mean intuitions fail. These contributions broaden the applicability of stochastic orders to heavy-tailed contexts with practical implications for risk, finance, and related fields.

Abstract

Many results on the convex order in the literature were stated for random variables with finite mean. For instance, a fundamental result in dependence modeling is that the sum of a pair of random random variables is upper bounded in convex order by that of its comonotonic version and lower bounded by that of its counter-monotonic version, and all existing proofs of this result require the random variables' expectations to be finite. We show that the above result remains true even when discarding the finite-mean assumption, and obtain several other results on the comparison of infinite-mean random variables via the convex order. To our surprise, we find two deceivingly similar definitions of the convex order, both of which exist widely in the literature, and they are not equivalent for random variables with infinite mean. This subtle discrepancy in definitions also applies to the supermodular order, and it gives rise to some incorrect statements, often found in the literature.

On convex order and supermodular order without finite mean

TL;DR

This paper extends the classical convex order and its relation to the supermodular order to random variables without finite means. It clarifies that two widely used definitions of convex order, cx and cx^dagger, coincide on finite-mean spaces but diverge when infinite means are allowed, with antisymmetry failing for cx^dagger in general. The authors prove that the fundamental two-variable bound involving counter-monotone and comonotone sums, X^{ct}+Y^{ct} ≤_cx X+Y ≤_cx X^{co}+Y^{co}, remains valid in the infinite-mean setting and extend analogous results to arbitrary dimensions. They also connect these results to optimal transport, provide ES-based and quantile-characterization conditions for infinite-mean convex order, and discuss multivariate nuances of the supermodular and concordance orders, highlighting cases where standard finite-mean intuitions fail. These contributions broaden the applicability of stochastic orders to heavy-tailed contexts with practical implications for risk, finance, and related fields.

Abstract

Many results on the convex order in the literature were stated for random variables with finite mean. For instance, a fundamental result in dependence modeling is that the sum of a pair of random random variables is upper bounded in convex order by that of its comonotonic version and lower bounded by that of its counter-monotonic version, and all existing proofs of this result require the random variables' expectations to be finite. We show that the above result remains true even when discarding the finite-mean assumption, and obtain several other results on the comparison of infinite-mean random variables via the convex order. To our surprise, we find two deceivingly similar definitions of the convex order, both of which exist widely in the literature, and they are not equivalent for random variables with infinite mean. This subtle discrepancy in definitions also applies to the supermodular order, and it gives rise to some incorrect statements, often found in the literature.

Paper Structure

This paper contains 6 sections, 16 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Two definitions of convex order
  3. Equivalent conditions for the convex order
  4. Convex order between sums of random variables
  5. Optimal transport and supermodular order
  6. Proofs of Theorems \ref{['th:ES-based-condition']}--\ref{['th:CX']}, \ref{['th:CXm-counter-monotonic']} and \ref{['th:counter-SM']}

Key Result

Proposition 1

On $L^1$, the two relations $\leqslant_{\rm cx}$ and $\leqslant^\dagger_{\rm cx}$ are equivalent. On $L^0$, $\leqslant_{\rm cx}$ implies $\leqslant^\dagger_{\rm cx}$, but the converse is not true.

Theorems & Definitions (39)

  • Example 1
  • Definition 1
  • Proposition 1
  • proof
  • Example 2
  • Example 3
  • Example 4
  • Proposition 2
  • proof
  • Theorem 1
  • ...and 29 more