An estimate for the probability of dependent events

Abstract

Abstract. In this note we prove an estimate for the probability that none of several events will occur provided that some of those events are dependent. This estimate (essentially due to Filaseta, Ford, Konyagin, Pomerance and Yu) can be applied to coverings of Z by systems of congruences, coverings of Z d by lattices and similar problems. Although this result is similar to the Lovász local lemma, it is independent of it. We will also prove a corollary in the style of the local lemma and show that in some situations our lower bound is stronger than that given by the Lovász lemma. As an illustration, we shall make some computations with an example considered earlier by Chen. IntroductionLet (Ω, F , P) be a probability space. Throughout, we shall use the notation E c = Ω\E for every event E. If the events E 1 , . . . , E ℓ are all independent of one another (more precisely, e.g., for every j = 2, . . . , ℓ, the events E j and ∪ j−1 i=1 E i are independent), thenThis implies that if P(E k ) < 1 for each k, then there is a positive probability that none of the events E 1 , . . . , E ℓ will occur. The Lovász local lemma (Erdős and Lovász, 1975) gives the same conclusion when the mutual independence of events is replaced by their 'rare' dependence. The 'symmetric' version of the local lemma states that if P(E k ) p for every k = 1, . . . , ℓ, where each of the events E k is independent of all other events except for at most d of them, then P(∩ ℓ k=1 E c k ) > 0 provided that ep(d + 1) 1. See, for instance, (Alon and Spencer, 2000), (Chen, 1997) for more precise and more general versions of this statement, although, in general, the constant e which occurs in this inequality cannot be replaced by any smaller constant (Shearer, 1985). There are many useful applications of the Lovász local lemma in combinatorics, graph theory, number theory as well as in other fields of mathematics, statistics and computer science (Czumaj and Scheideler, 2000), (Deng, Stinson and Wei, 2004), (Dubickas, 2008), (Grytczuk, 2007), (Scott and Sokal, 2006), (Srinivasan, 2006), (Szabó, 1990).Recently, in a paper of Filaseta, Ford, Konyagin, Pomerance and Yu (2007) devoted to the study of covering systems of congruences, another lower bound for the quantity P(∩ ℓ k=1 E c k ), where some of the events E i and E j are dependent, was obtained. Although 2000 Mathematics Subject Classification. 60A99, 60C05, 11B25.

Extremal independence in discrete random systems

“…The best-known lower bounds are Lovász Local Lemma [9] (LLL), its various extensions [42] and the mentioned Suen's inequality [41]. Also, Dubickas [8] proved another lower bound that holds under weaker assumptions than LLL. Notice that Janson's and Dubickas' inequalities work only in the case when pairwise indenpendce implies mutual independence as it often happens in random subsets settings when the events are, so-called, up-sets.…”

“…Lovász Local Lemma (LLL) [9] is the best-known lower bound and has various extensions; see, for example, [42] and references therein. Recently, Dubickas [8] gives another lower bound which holds under weaker assumptions than LLL. The dependencies among events in all aforementioned results are represented by a certain combinatorial structure called dependency graph.…”