Upper Tail Estimates with Combinatorial Proofs

TL;DR

This paper develops simplified combinatorial proofs for upper tail bounds in probability, applying them to expander walks, polynomials with dependent variables, and subgraph counts in random graphs, improving understanding of concentration phenomena.

Contribution

It introduces new, simpler combinatorial proofs for upper tail bounds applicable to various probabilistic models, including expander walks, dependent polynomials, and Erdős-Rényi graphs.

Findings

Proved a concentration bound for expander random walks that is nearly optimal.

Derived an upper tail bound for polynomials with dependent variables.

Matched existing bounds for subgraph counts in Erdős-Rényi graphs.

Abstract

We study generalisations of a simple, combinatorial proof of a Chernoff bound similar to the one by Impagliazzo and Kabanets (RANDOM, 2010). In particular, we prove a randomized version of the hitting property of expander random walks and apply it to obtain a concentration bound for expander random walks which is essentially optimal for small deviations and a large number of steps. At the same time, we present a simpler proof that still yields a "right" bound settling a question asked by Impagliazzo and Kabanets. Next, we obtain a simple upper tail bound for polynomials with input variables in $[0, 1]$ which are not necessarily independent, but obey a certain condition inspired by Impagliazzo and Kabanets. The resulting bound is used by Holenstein and Sinha (FOCS, 2012) in the proof of a lower bound for the number of calls in a black-box construction of a pseudorandom generator from a one-way function. We then show that the same technique yields the upper tail bound for the number of copies of a fixed graph in an Erd\H{o}s-R\'enyi random graph, matching the one given by Janson, Oleszkiewicz and Ruci\'nski (Israel J. Math, 2002).