  The well-known "Janson's inequality" gives Poisson-like upper bounds for the
lower tail probability \Pr(X \le (1-\eps)\E X) 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 \eps=1) of Janson, Luczak and
Rucinski.
