The infamous upper tail

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The infamous upper tail
Let be a finite index set and k 1 a given integer. Let further S []k be an arbitrary family of k element subsets of . Consider a (binomial) random subset p of , where p = (pi : i ) and a random variable X counting the elements of S that are contained in this random subset. In this paper we survey techniques of obtaining upper bounds on the upper tail probabilities P(X + t) for t > 0. Seven techniques, ranging from Azuma's inequality to the purely combinatorial deletion method, are described, illustrated and compared against each other for a couple of typical applications. As one application, we obtain essentially optimal bounds for the upper tails for the numbers of subgraphs isomorphic to K4 or C4 in a random graph G(n, p), for certain ranges of p.
Svante Janson, Andrzej Rucinski
Added 23 Dec 2010
Updated 23 Dec 2010
Type Journal
Year 2002
Where RSA
Authors Svante Janson, Andrzej Rucinski
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