The Janson inequalities for general up-sets

TL;DR

This paper extends the Janson inequalities from principal up-sets to arbitrary up-sets, broadening their applicability in probability theory by leveraging positive correlation properties.

Contribution

It generalizes existing inequalities for the lower tail of up-set distributions to all up-sets, removing previous restrictions on their structure.

Findings

Inequalities hold for all up-sets, not just principal ones.

Proofs are adapted to rely solely on positive correlation.

Results apply to a broader class of events in probabilistic analysis.

Abstract

Janson and Janson, Luczak and Rucinski proved several inequalities for the lower tail of the distribution of the number of events that hold, when all the events are up-sets (increasing events) of a special form - each event is the intersection of some subset of a single set of independent events (i.e., a principal up-set). We show that these inequalities in fact hold for arbitrary up-sets, by modifying existing proofs to use only positive correlation, avoiding the need to assume positive correlation conditioned on one of the events.