Concentration inequalities for functions of independent variables

Abstract

Following the entropy method this paper presents general concentration inequalities, which can be applied to combinatorial optimization and empirical processes. The inequalities give improved concentration results for optimal travelling salesmen tours, Steiner trees and the eigenvalues of random symmetric matrices.

Concentration bounds for geometric Poisson functionals: Logarithmic Sobolev inequalities revisited

“…Here one should notice that according to Corollary 3.6, the condition V + ≤ aF guarantees E|F | < ∞. It was pointed out in [27] that for any λ < 0,…”

Concentration for self‐bounding functions and an inequality of Talagrand

“…A referee pointed out that the case a = 1 was dealt with by Devroye in [3]. Another referee pointed out that the case b = 0 is dealt with independently by Maurer in [9], using quite different techniques (see Theorem 13 of that paper). Now we consider an inequality of Talagrand (first formulated by S. Janson, E. Shamir, M. Steele, and J. Spencer, as noted in [12]), which is particularly useful in discrete mathematics and theoretical computer science; see also [14,15].…”

Concentration for Poisson functionals: Component counts in random geometric graphs

“…The proof of Theorem 3.2 below is very similar to the proof of [2, Theorem 3.10] which in turn is an adaptation of the proof of [22,Theorem 13] for Poisson functionals. For the sake of completeness, we carry out the modified argumentation.…”