Concentration inequalities on the multislice and for sampling without replacement

TL;DR

This paper develops concentration inequalities on the multislice and for sampling without replacement, providing bounds for convex functions, polynomials, and applications to Erdős–Rényi graphs, along with proofs of key inequalities.

Contribution

It introduces new concentration bounds on the multislice based on log-Sobolev inequalities and extends these results to sampling without replacement scenarios.

Findings

Concentration bounds for convex functions and multilinear polynomials on the multislice.

Application of concentration results to the triangle count in Erdős–Rényi graphs.

Simplified proof of Serfling's inequality with a slightly weaker exponent.

Abstract

We present concentration inequalities on the multislice which are based on (modified) log-Sobolev inequalities. This includes bounds for convex functions and multilinear polynomials. As an application we show concentration results for the triangle count in the $G(n,M)$ Erd\H{o}s--R\'{e}nyi model resembling known bounds in the $G(n,p)$ case. Moreover, we give a proof of Talagrand's convex distance inequality for the multislice. Interpreting the multislice in a sampling without replacement context, we furthermore present concentration results for $n$ out of $N$ sampling without replacement. Based on a bounded difference inequality involving the finite-sampling correction factor $1- n/N$, we present an easy proof of Serfling's inequality with a slightly worse factor in the exponent, as well as a sub-Gaussian right tail for the Kolmogorov distance between the empirical measure and the true distribution of the sample.