Halving the bounds for the Markov, Chebyshev, and Chernoff Inequalities using smoothing

TL;DR

This paper introduces a simple smoothing technique that halves the bounds of the Markov, Chebyshev, and Chernoff inequalities, improving their tightness in tail probability estimation.

Contribution

The paper presents a novel smoothing method that reduces the bounds of three fundamental inequalities, often without auxiliary randomness, enhancing their precision.

Findings

Bounds can be halved using smoothing techniques.

In many cases, auxiliary randomness is unnecessary for halving bounds.

The method improves tail probability bounds for various distributions.

Abstract

The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equality. Here we will show that through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.