For increasing containment events in a Bernoulli product space, let , , andwhere the sum is ordered and dependency means overlapping coordinate supports. ThenIn particular it is at most when , and at most when .
The Janson dependency sum adds over ordered pairs of distinct containment events whose coordinate supports overlap.
For containment-event indicators in a Bernoulli product space,It follows by exposing the avoidance events in sequence, separating disjoint coordinate supports, and applying Harris' inequality to the remaining decreasing events.
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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.