The infamous upper tail

Abstract

Let ⌫ be a finite index set and k Ն 1 a given integer. Let further ʕ [⌫]Յk be an arbitrary family of k element subsets of ⌫. Consider a (binomial) random subset ⌫ p of ⌫, where p ϭ ( p i : i ʦ ⌫) and a random variable X counting the elements of that are contained in this random subset. In this paper we survey techniques of obtaining upper bounds on the upper tail probabilities ‫(ސ‬X Ն ϩ t) for t Ͼ 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 K 4 or C 4 in a random graph G(n, p), for certain ranges of p.

Large deviations for sums of partly dependent random variables

“…The idea to use the Hajnal-Szemerédi Theorem in this context is due to Ruciński [16,18] and Pemmaraju [20] (independently). Pemmaraju [20] further explores the possibility of improving the bound by finding by other means, for specific Γ, a partition of Γ into fewer than ∆ 1 independent sets of (almost) the same size.…”

“…Hoeffding's method for this case is based on breaking up the sum (1.2) into several parts, each part being a sum of independent variables. The same idea has been used in a somewhat different form (see Remark 5.2) by, among others, [21], [18], [20]. We will show how Hoeffding's original method, with only minor modifications, extends to general sums (1.2) and (1.1).…”

“…See also partial results and discussions in [22] and [18]; the present method is essential the same as "Break-up" in the latter. (It is a slight improvement, see Remark 5.2.…”

“…In the case of independent summands, some of the most important contributions are Bernstein [4], Cramér [9], Feller [10], Chernoff [8], Okamoto [19], Bennett [3], and Hoeffding [14]. For dependent summands, there are many results proved by many authors using different methods under various assumptions; for a few of these, see the surveys in [16] and [18].…”

Equitable coloring of random graphs

“…Applications of the Hajnal-Szemerédi theorem and recent results on equitable colorings of graphs can be found in (among others) [1], [2], [9], [11], [12], [19]. Equitable coloring turned out to be useful in establishing bounds on tails of sums of dependent variables [6], [8], [18].…”