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Upper Tail Estimates with Combinatorial Proofs

Authors Jan Hazla, Thomas Holenstein



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Jan Hazla
Thomas Holenstein

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Jan Hazla and Thomas Holenstein. Upper Tail Estimates with Combinatorial Proofs. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 392-405, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)
https://doi.org/10.4230/LIPIcs.STACS.2015.392

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 use it to obtain an optimal expander random walk concentration bound settling a question asked by Impagliazzo and Kabanets. Next, we obtain an 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 applied 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 also show that the same technique yields the upper tail bound for the number of copies of a fixed graph in an Erdös–Rényi random graph, matching the one given by Janson, Oleszkiewicz, and Rucinski (Israel J. Math, 2002).
Keywords
  • concentration bounds
  • expander random walks
  • polynomial concentration

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