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Convex comparison of Gaussian mixtures

Benjamin Jourdain, Gilles Pagès

Abstract

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. We give and discuss intrinsic necessary and sufficient conditions for convex ordering. On the examples that we have worked out, the two conditions appear to be closely related.

Convex comparison of Gaussian mixtures

Abstract

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order between a Gaussian distribution and a Gaussian mixture. We give and discuss intrinsic necessary and sufficient conditions for convex ordering. On the examples that we have worked out, the two conditions appear to be closely related.

Paper Structure

This paper contains 6 sections, 10 theorems, 58 equations.

Table of Contents

  1. Convex ordering between a Gaussian distribution and a Gaussian mixture
  2. Main results and comments
  3. Proofs of the main results
  4. Examples
  5. Comparison of the sufficient conditions with \ref{['condconvjp']}
  6. The case with $n=2$ Gaussian distributions in the mixture

Key Result

Theorem 1.1

Let $n\ge 2$, $\Sigma,\Sigma_1,\cdots,\Sigma_n\in{\cal S}_+(d)$ and $(p_1,\cdots,p_n)\in(0,1)^n$ such that $\sum_{i=1}^n p_i=1$. Then implies which implies which in turn implies Moreover, the four conditions correl, inecov, compccgg and inegsqrt are equivalent

Theorems & Definitions (28)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 18 more