Borel–Cantelli Lemmas
Summable event probabilities force only finitely many occurrences almost surely. With independence, a divergent sum gives infinitely many occurrences; the repeated-event example explains why that hypothesis cannot simply be omitted.
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Probability Space · Independent Events · Almost Sure Convergence
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Start here: Notes on Probability
Greg Lawler · Lecture notes · English · Free full text
§4 Sums of independent random variables, Lemma 4.3 (Borel-Cantelli Lemma) and the dependent-event counterexample
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Probability Theory in the library · Preparation and skip guidance