 Janson inequality

For increasing containment events $A_i$ in a Bernoulli product space, let $X={\sum }_i{1}_{A_i}$, $\mu =\mathbb{E}X$, andwhere the sum is ordered and dependency means overlapping coordinate supports. ThenIn particular it is at most $e^{-\mu /2}$ when $\Delta \le \mu$, and at most $e^{-{\mu }^2/(2\Delta )}$ when $\Delta \ge \mu$.

The Janson inequality is a result in probability theory, particularly in the context of the study of random variables and dependent events. It provides a bound on the probability that a sum of random variables exceeds its expected value. Specifically, it is often used when dealing with random variables that exhibit some form of dependence.