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, who gave statement to the lemma in the first decades of the 20th century.
The lemma states that, under certain conditions, an event will have probability of either zero or one.
Accordingly, it is the best-known of a class of similar theorems, known as zero-one laws.
Let E1, E2, ... be a sequence of events in some probability space.
The Borel–Cantelli lemma states:[3][4] Borel–Cantelli lemma — If the sum of the probabilities of the events {En} is finite
Here, "lim sup" denotes limit supremum of the sequence of events, and each event is a set of outcomes.
That is, lim sup En is the set of outcomes that occur infinitely many times within the infinite sequence of events (En).
The theorem therefore asserts that if the sum of the probabilities of the events En 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.
Suppose (Xn) is a sequence of random variables with Pr(Xn = 0) = 1/n2 for each n. The probability that Xn = 0 occurs for infinitely many n is equivalent to the probability of the intersection of infinitely many [Xn = 0] events.
The intersection of infinitely many such events is a set of outcomes common to all of them.
However, the sum ΣPr(Xn = 0) converges to π2/6 ≈ 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 Xn = 0 occurring for infinitely many n is 0.
Almost surely (i.e., with probability 1), Xn is nonzero for all but finitely many n. Let (En) be a sequence of events in some probability space.
[5] For general measure spaces, the Borel–Cantelli lemma takes the following form: Borel–Cantelli Lemma for measure spaces — Let μ be a (positive) measure on a set X, with σ-algebra F, and let (An) be a sequence in F. If
The lemma states: If the events En are independent and the sum of the probabilities of the En diverges to infinity, then the probability that infinitely many of them occur is 1.
The assumption of independence can be weakened to pairwise independence, but in that case the proof is more difficult.
The infinite monkey theorem follows from this second lemma.
The lemma can be applied to give a covering theorem in Rn.
Specifically (Stein 1993, Lemma X.2.1), if Ej is a collection of Lebesgue measurable subsets of a compact set in Rn such that
It is sufficient to show the event that the En's did not occur for infinitely many values of n has probability 0.
Another related result is the so-called counterpart of the Borel–Cantelli lemma.
It is a counterpart of the Lemma in the sense that it gives a necessary and sufficient condition for the limsup to be 1 by replacing the independence assumption by the completely different assumption that
is monotone increasing for sufficiently large indices.
occurs) is one if and only if there exists a strictly increasing sequence of positive integers
This simple result can be useful in problems such as for instance those involving hitting probabilities for stochastic process with the choice of the sequence
Now, notice that by the Cauchy-Schwarz Inequality,
But the left side is precisely the probability that the