Borel-Cantelli lemma

related topics
{math, number, function}
{rate, high, increase}
{day, year, event}
{country, population, people}

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.


Statement of lemma for probability spaces

Let (En) 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 En is the set of outcomes that occur infinitely many times within the infinite sequence of events (En). Explicitly,

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.


For example, 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 ∑Pr(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.

Proof [

Let (En) be a sequence of events in some probability space and suppose that the sum of the probabilities of the En is finite. That is suppose:

Full article ▸

related documents
Double negative elimination
Weierstrass–Casorati theorem
Hash collision
Nowhere dense set
Magma computer algebra system
Category (mathematics)
Dyadic rational
Partial function
Calculus with polynomials
Stirling number
Algebraic number
T1 space
Cayley's theorem
Malleability (cryptography)
Real line
Regular space
Steiner system
Alternative algebra
Partial fractions in integration
Byte-order mark
Atlas Autocode
Algebraic extension
Residue (complex analysis)
GNU Octave
Euler's criterion
Decision problem