2013 IEEE 54th Annual Symposium on Foundations of Computer Science (2009)
Oct. 25, 2009 to Oct. 27, 2009
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/FOCS.2009.62
We prove a simple concentration inequality, which is an extension of the Chernoff bound and Hoeffding's inequality for binary random variables. Instead of assuming independence of the variables we use a slightly weaker condition, namely bounds on the co-moments. This inequality allows us to simplify and strengthen several known direct-product theorems and establish new threshold direct-product theorems. Threshold direct-product theorems are statements of the following form: If one instance of a problem can be solved with probability at most p, then solving significantly more than a p-fraction among multiple instances has negligible probability. Results of this kind are crucial when distinguishing whether a process succeeds with probability s or c, for 0<s<c<1. Here standard direct-product theorems are of no help since even a process which can solve one instance with probability c will only be able to solve all k instances withe exponentially small probability.
Concentration inequality, Chernoff bound, XOR Lemma, Direct-product Theorem
Falk Unger, "A Probabilistic Inequality with Applications to Threshold Direct-Product Theorems", 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, vol. 00, no. , pp. 221-229, 2009, doi:10.1109/FOCS.2009.62