In probability theory, the Borel–Cantelli lemma is a theorem about sequences of events. In general, it is a result in measure theory. It is named after Émile Borel and Francesco Paolo Cantelli. A related result, sometimes called the second Borel–Cantelli lemma, is a partial converse of the first Borel–Cantelli lemma.
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Statement of lemma for probability spaces
Let (E_{n}) be a sequence of events in some probability space. The Borel–Cantelli lemma states:
Here, "lim sup" denotes limit superior of the sequence of events, and each event is a set of outcomes. That is, lim sup E_{n} is the set of outcomes that occur infinitely many times within the infinite sequence of events (E_{n}). Explicitly,
The theorem therefore asserts that if the sum of the probabilities of the events E_{n} is finite, then the set of all outcomes that are "repeated" infinitely many times must occur with probability zero. Note that no assumption of independence is required.
Example
For example, suppose (X_{n}) is a sequence of random variables with Pr(X_{n} = 0) = 1/n^{2} for each n. The probability that X_{n} = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [X_{n} = 0] events. The intersection of infinitely many such events is a set of outcomes common to all of them. However, the sum ∑Pr(X_{n} = 0) converges to π^{2}/6 ≈ 1.645 < ∞, and so the Borel–Cantelli Lemma states that the set of outcomes that are common to infinitely many such events occurs with probability zero. Hence, the probability of X_{n} = 0 occurring for infinitely many n is 0. Almost surely (i.e., with probability 1), X_{n} is nonzero for all but finitely many n.
Proof ^{[}
Let (E_{n}) be a sequence of events in some probability space and suppose that the sum of the probabilities of the E_{n} is finite. That is suppose:
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