Two new Probability inequalities and Concentration Results

TL;DR

This paper introduces two new probability inequalities that extend martingale inequalities, enabling better analysis of combinatorial problems under complex distributions, with broad applications including TSP, MWST, graph coloring, and more.

Contribution

The paper presents two novel probability inequalities that generalize existing martingale inequalities, applicable to heavy-tailed and inhomogeneous distributions in combinatorial analysis.

Findings

Proved inequalities applicable to diverse distributions.

Extended concentration results beyond i.i.d. settings.

Applied to multiple combinatorial and geometric problems.

Abstract

Concentration results and probabilistic analysis for combinatorial problems like the TSP, MWST, graph coloring have received much attention, but generally, for i.i.d. samples (i.i.d. points in the unit square for the TSP, for example). Here, we prove two probability inequalities which generalize and strengthen Martingale inequalities. The inequalities provide the tools to deal with more general heavy-tailed and inhomogeneous distributions for combinatorial problems. We prove a wide range of applications - in addition to the TSP, MWST, graph coloring, we also prove more general results than known previously for concentration in bin-packing, sub-graph counts, Johnson-Lindenstrauss random projection theorem. It is hoped that the strength of the inequalities will serve many more purposes.