An algorithmic proof of the Chernoff-Hoeffding bound

SERIE B - INFORMATIK Algorithmic Chernoff-Hoeffding Inequalities in Integer Programming

The Converse Part of The Theorem for Quantum Hoeffding Bound

We prove the converse part of the theorem for quantum Hoeffding bound on the asymptotics of quantum hypothesis testing, essentially based on an argument developed by Nussbaum and Szkola in proving the converse part of the quantum Chernoff bound. Our result complements Hayashi’s proof of the direct (achievability) part of the theorem, so that the quantum Hoeffding bound has now been established.

Constructive Proofs of Concentration Bounds

We give a simple combinatorial proof of the Chernoff-Hoeffding concentration bound [Che52, Hoe63], which says that the sum of independent {0, 1}-valued random variables is highly concentrated around the expected value. Unlike the standard proofs, our proof does not use the method of higher moments, but rather uses a simple and intuitive counting argument. In addition, our proof is constructive ...

Error Exponents in Quantum Hypothesis Testing

In the simple quantum hypothesis testing problem, upper bounds on the error probabilities are shown based on a key matrix inequality. Concerning the error exponents, the upper bounds lead to noncommutative analogues of the Chernoff bound and the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Ou...

A note on the distribution of the number of prime factors of the integers

The Chernoff-Hoeffding bounds are fundamental probabilistic tools. An elementary approach is presented to obtain a Chernoff-type upper-tail bound for the number of prime factors of a random integer in {1, 2, . . . , n}. The method illustrates tail bounds in negatively-correlated settings.