The convex distance inequality for dependent random variables, with applications to the stochastic travelling salesman and other problems

TL;DR

This paper establishes concentration inequalities for dependent random variables under the Dobrushin condition, extending Talagrand's convex distance inequality and applying it to combinatorial optimization and statistical physics problems.

Contribution

It generalizes Talagrand's convex distance inequality to weakly dependent variables satisfying the Dobrushin condition, with applications to various complex stochastic problems.

Findings

Proved Talagrand's convex distance inequality for dependent variables.

Derived concentration bounds for the stochastic traveling salesman problem.

Applied results to the Steiner tree, Curie-Weiss model, and exponential random graphs.

Abstract

We prove concentration inequalities for general functions of weakly dependent random variables satisfying the Dobrushin condition. In particular, we show Talagrand's convex distance inequality for this type of dependence. We apply our bounds to a version of the stochastic salesman problem, the Steiner tree problem, the total magnetisation of the Curie-Weiss model with external field, and exponential random graph models. Our proof uses the exchangeable pair method for proving concentration inequalities introduced by Chatterjee (2005). Another key ingredient of the proof is a subclass of $(a,b)$-self-bounding functions, introduced by Boucheron, Lugosi and Massart (2009).