The Borel-Cantelli Lemmas for contaminated events, and small maxima

Abstract: For a sequence of independent events $E_n$ the sum of the associated zero-one random variables $1_{E_n}$ is almost surely finite or almost surely infinite according as the sum of the probabilities converges or diverges. In this paper the events $E_n$ are contaminated. What can one say about $\sum1_{D_n}$ when $D_n=E_n\setminus A_n$ for a sequence of events $A_n$ with vanishing probability? The behaviour depends on the relation between the events $E_n$ and $A_n$ and on the size of the events. We prove a Borel-Cantelli lemma for the contaminated variables and give an application in extreme value theory.