License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.STACS.2015.392
URN: urn:nbn:de:0030-drops-49291
URL: https://drops.dagstuhl.de/opus/volltexte/2015/4929/
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Hazla, Jan ; Holenstein, Thomas

Upper Tail Estimates with Combinatorial Proofs

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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).

BibTeX - Entry

@InProceedings{hazla_et_al:LIPIcs:2015:4929,
  author =	{Jan Hazla and Thomas Holenstein},
  title =	{{Upper Tail Estimates with Combinatorial Proofs}},
  booktitle =	{32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)},
  pages =	{392--405},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-78-1},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{30},
  editor =	{Ernst W. Mayr and Nicolas Ollinger},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/4929},
  URN =		{urn:nbn:de:0030-drops-49291},
  doi =		{10.4230/LIPIcs.STACS.2015.392},
  annote =	{Keywords: concentration bounds, expander random walks, polynomial concentration}
}

Keywords: concentration bounds, expander random walks, polynomial concentration
Collection: 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)
Issue Date: 2015
Date of publication: 26.02.2015


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