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

2017/07/06 by Balkema, Guus
#60E15 #60F15 #60F20 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1707.01678

Abstract

For a sequence of independent events E_n the sum of the associated zero-one random variables 1_En 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 ∑1_Dn when D_n=E_n∖ 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.

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