The lower tail: Poisson approximation revisited

The well‐known “Janson's inequality” gives Poisson‐like upper bounds for the lower tail probability when X is the sum of dependent indicator random variables of a special form. We show that, for large deviations, this inequality is optimal whenever X is approximately Poisson, i.e., when the dependencies are weak. We also present correlation‐based approaches that, in certain symmetric applications, yield related conclusions when X is no longer close to Poisson. As an illustration we, e.g., consider subgraph counts in random graphs, and obtain new lower tail estimates, extending earlier work (for the special case ) of Janson, ?uczak and Ruciński. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 219–246, 2016