Generic and lifted probabilistic comparisons -- max replaces minmax

TL;DR

This paper introduces a unified framework for probabilistic comparison results, including a novel lifting procedure that enhances existing methods, supported by numerical experiments and connections to classical principles like Slepian's max.

Contribution

It presents a new comparison framework with a lifting procedure that improves upon traditional methods, unifies classical results, and broadens application scope.

Findings

Lifting procedure offers a substantial upgrade over general comparison strategies.

Many classical comparison results are special cases of the new framework.

Numerical experiments strongly support the theoretical predictions.

Abstract

In this paper we introduce a collection of powerful statistical comparison results. We first present the results that we obtained while developing a general comparison concept. After that we introduce a separate lifting procedure that is a comparison concept on its own. We then show how in certain scenarios the lifting procedure basically represents a substantial upgrade over the general strategy. We complement the introduced results with a fairly large collection of numerical experiments that are in an overwhelming agreement with what the theory predicts. We also show how many well known comparison results (e.g. Slepian's max and Gordon's minmax principle) can be obtained as special cases. Moreover, it turns out that the minmax principle can be viewed as a single max principle as well. The range of applications is enormous. It starts with revisiting many of the results we created in recent years in various mathematical fields and recognizing that they are fully self-contained as their starting blocks are specialized variants of the concepts introduced here. Further upgrades relate to core comparison extensions on the one side and more practically oriented modifications on the other. Those that we deem the most important we discuss in several separate companion papers to ensure preserving the introductory elegance and simplicity of what is presented here.