Large deviation upper bounds for sums of positively associated indicators

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We give exponential upper bounds for P(S ≤ k), in particular P(S=0), where S is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give examples in which we compare with a famous exponential inequality for sums of correlated indicators, the Janson inequality. Here our bound sometimes proves to be superior to Janson's bound.