New versions of Suen's correlation inequality

1. Introduction Suen 8] found a remarkable correlation inequality, giving estimates for the probability that a collection of dependent random indicator variables vanish simultaneously, or in other words, for the probability that none of a collection of dependent events occurs. The present author 4, 3] has found similar inequalities for a much more restricted situation; when applicable, these inequalities are somewhat better than Suen's, although the diierence is negligible in many cases. (See Section 8 below.) Those inequalities have been used by several diierent authors for a variety of problems; there are, however, many situations where they are not applicable (see 8, 5] for two examples) and then Suen's inequality is a very attractive choice. The purpose of the present note is to present some improvements and mod-iications of Suen's original inequality which (we hope) will be easy to apply in diierent situations. The estimates considered here are exponential (unlike for example Cheby-shev's inequality), in the sense that they typically are similar to the estimate exp() for the independent case, where is the expected number of events. They are thus aimed at the case when the studied probability is very small, and has to be shown to be very small. In many applications, constants oc-curing in the estimates, even in the exponents, are immaterial; on the other hand, there are applications where very precise estimates are desired. For this reason, and because diierent versions of the inequality turn out to be useful in diierent situations, we will give several diierent versions of our estimates. We give several upper bounds to the probability of simultaneous vanishing of a collection of indicator variables in Section 3; these are perhaps the main results of the paper. We give some corresponding lower bounds in Section 4, and in Section 5 an upper bound for the probability that only a few of the variables are non-zero. Section 6 contains the proofs of the results, while Section 7 contains three examples related to the sharpness of the results. Finally, Section 8 contains a short discussion of the results and some open problems. Acknowledgement. This paper has beneetted from discussions with par