# Borel–Cantelli lemma

The **Borel–Cantelli lemma** is a result in probability theory that relates the convergence or divergence of the series of event probabilities \( \sum_{n} \mathbb{P}(A_n) \) to whether infinitely many of the events \( A_1, A_2, \ldots \) occur. In its first form it states that if the sum of the probabilities converges, then, with probability 1, only finitely many of the events occur; in its second form, for mutually independent events with a divergent sum, infinitely many of the events occur with probability 1. The result was introduced by [Émile Borel](https://www.edgepedia.org/emile-borel) in 1909 as his Zero-One Law and refined by [Francesco Paolo Cantelli](https://www.edgepedia.org/francesco-paolo-cantelli), who observed that the first form needs no independence assumption.<sup>[1](https://www.math.purdue.edu/~eremenko/Pdf/barone.pdf)</sup><sup> • </sup><sup>[2](https://arxiv.org/abs/0710.4264)</sup>

| Key fact | Detail |
|---|---|
| First lemma | If \( \sum_{n=1}^{\infty} \mathbb{P}(A_n) < \infty \), then \( \mathbb{P}(A_n \text{ i.o.}) = 0 \); no independence is required<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)</sup> |
| Second lemma | If the \( A_n \) are mutually independent and \( \sum_{n=1}^{\infty} \mathbb{P}(A_n) = \infty \), then \( \mathbb{P}(A_n \text{ i.o.}) = 1 \)<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup> |
| Zero-one corollary | For mutually independent events, \( \mathbb{P}(A_n \text{ i.o.}) \) is either 0 or 1, according to whether the probability sum converges or diverges<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup> |
| Weakened independence | Erdős and Rényi showed the second lemma holds under pairwise independence; the Kochen–Stone lemma gives a lower bound \( \mathbb{P}(A_n \text{ i.o.}) \geq \alpha \) without independence<sup>[5](https://arxiv.org/abs/2004.11324)</sup><sup> • </sup><sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup>  |
| Main use | Proving almost sure convergence of random variables and the strong law of large numbers<sup>[6](https://tsoo-math.github.io/ucl/borel.html)</sup> |
| History | Borel (1909) proved the result assuming total independence; Cantelli removed that assumption in the convergent case<sup>[2](https://arxiv.org/abs/0710.4264)</sup> |

## The limsup of a sequence of events

For a sequence of events \( A_1, A_2, \ldots \) on a probability space \( (\Omega, \mathcal{F}, \mathbb{P}) \), the relevant object is the limit superior of the sets:

\[ \limsup_{n \to \infty} A_n = \bigcap_{m \geq 1} \bigcup_{n \geq m} A_n. \]

This is the event consisting of outcomes \( \omega \) that belong to infinitely many of the \( A_n \), written \( \{A_n \text{ i.o.}\} \), where "i.o." stands for "infinitely often".<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)</sup><sup> • </sup><sup>[6](https://tsoo-math.github.io/ucl/borel.html)</sup>  The dual limit inferior,

\[ \liminf_{n \to \infty} A_n = \bigcup_{m \geq 1} \bigcap_{n \geq m} A_n, \]

is the event that, from some index onward, all the \( A_n \) occur. Fatou's lemma for sets links the two to the probabilities of the individual events:

\[ \mathbb{P}\left(\liminf_{n \to \infty} A_n\right) \leq \liminf_{n \to \infty} \mathbb{P}(A_n) \leq \limsup_{n \to \infty} \mathbb{P}(A_n) \leq \mathbb{P}\left(\limsup_{n \to \infty} A_n\right). \]

<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)</sup>

## The first lemma

**Borel–Cantelli I.** If \( \sum_{n=1}^{\infty} \mathbb{P}(A_n) < \infty \), then \( \mathbb{P}(A_n \text{ i.o.}) = 0 \).<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)</sup>

The proof is short. For every \( N \), the event \( \{A_n \text{ i.o.}\} \) is contained in \( \bigcup_{n \geq N} A_n \), so by subadditivity

\[ \mathbb{P}(A_n \text{ i.o.}) \leq \mathbb{P}\left(\bigcup_{n \geq N} A_n\right) \leq \sum_{n \geq N} \mathbb{P}(A_n). \]

Since the series converges, its tail tends to zero, and letting \( N \to \infty \) gives the result.<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)</sup> No independence assumption of any kind is needed, and no other hypothesis either.<sup>[6](https://tsoo-math.github.io/ucl/borel.html)</sup><sup> • </sup><sup>[7](https://learn.thuztra.com/reference/independence-and-the-borel-cantelli-lemmas)</sup>

A typical application shows that a 0–1 valued sequence \( X_n \) with \( \sum_{n} \mathbb{P}(X_n = 1) < \infty \) converges almost surely to 0: the event \( \{X_n = 1 \text{ i.o.}\} \) is exactly the event that \( X_n \) fails to converge to 0, and the first lemma makes it a null event.<sup>[8](https://www.math.uni.wroc.pl/~szekli/documents/m-CHAINS-21/B.pdf)</sup>

## The second lemma and the zero-one law

**Borel–Cantelli II.** If the events \( A_n \) are mutually independent and \( \sum_{n=1}^{\infty} \mathbb{P}(A_n) = \infty \), then \( \mathbb{P}(A_n \text{ i.o.}) = 1 \).<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup>

The proof uses independence and the inequality \( 1 - x \leq e^{-x} \) for \( x \geq 0 \):

\[ \mathbb{P}\left(\bigcap_{\ell=m}^{n} A_\ell^{c}\right) = \prod_{\ell=m}^{n} (1 - \mathbb{P}(A_\ell)) \leq \exp\left(-\sum_{\ell=m}^{n} \mathbb{P}(A_\ell)\right). \]

Because the series diverges, the right side tends to 0 as \( n \to \infty \) for every fixed \( m \); by continuity from above, almost every outcome lies in infinitely many of the events.<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup><sup> • </sup><sup>[9](https://www.sciencedirect.com/topics/mathematics/borel-cantelli-lemma)</sup>

Together the two lemmas give a zero-one law: for mutually independent events, \( \mathbb{P}(A_n \text{ i.o.}) \) is either 0 or 1, depending only on whether \( \sum_n \mathbb{P}(A_n) \) converges or diverges.<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup><sup> • </sup><sup>[10](https://crash-statistics.netlify.app/chapters/ch1-probability/ch4-independence)</sup>

## Why independence matters in the divergent case

Divergence of \( \sum_n \mathbb{P}(A_n) \) alone does not force infinitely many occurrences. On \( (0, 1] \) with uniform measure, the nested events \( A_n = (0, 1/n] \) satisfy \( \sum_n \mathbb{P}(A_n) = \sum_n 1/n = \infty \), yet \( \{A_n \text{ i.o.}\} = \emptyset \). In general the probability of \( \{A_n \text{ i.o.}\} \) depends on the relations among the events.<sup>[11](https://web.stanford.edu/~montanar/TEACHING/Stat310B/NOTES/amir_notes.pdf)</sup>

The independence hypothesis can be weakened substantially. Erdős and Rényi proved in 1959 that pairwise independence suffices for the conclusion \( \mathbb{P}(A_n \text{ i.o.}) = 1 \), and the second lemma also holds if the events are pairwise negative quadrant dependent, meaning \( \mathbb{P}(A_i \cap A_j) \leq \mathbb{P}(A_i)\mathbb{P}(A_j) \) for all \( i \neq j \).<sup>[5](https://arxiv.org/abs/2004.11324)</sup><sup> • </sup><sup>[12](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/Chandra%20T.K.%20The%20Borel-Cantelli%20lemma%20%28Springer%2C%202012%29%28ISBN%209788132206767%29%28O%29%28114s%29_MV_.pdf)</sup>  The Kochen–Stone lemma generalizes Borel–Cantelli II further: if \( \sum_k \mathbb{P}(A_k) = \infty \) and

\[ \limsup_{n \to \infty} \frac{\left(\sum_{k=1}^{n} \mathbb{P}(A_k)\right)^{2}}{\sum_{1 \leq j,k \leq n} \mathbb{P}(A_j \cap A_k)} = \alpha > 0, \]

then \( \mathbb{P}(A_n \text{ i.o.}) \geq \alpha \).<sup>[4](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup> Later work has mapped a whole lattice of weak-independence conditions, including nonpositive correlation of the indicator variables, that imply the converse conclusion.<sup>[5](https://arxiv.org/abs/2004.11324)</sup>

## Applications

**Almost sure convergence.** A random sequence \( X_n \) converges almost surely to \( X \) if and only if \( \mathbb{P}(|X_n - X| > \varepsilon \text{ i.o.}) = 0 \) for every \( \varepsilon > 0 \), so the first lemma converts summability of tail probabilities into almost sure convergence.<sup>[6](https://tsoo-math.github.io/ucl/borel.html)</sup> The same mechanism proves a strong law of large numbers under a fourth-moment assumption: for i.i.d. \( X_i \) with \( \mathbb{E}X_1 = 0 \) and \( \mathbb{E}X_1^4 < \infty \), [Markov's inequality](https://www.edgepedia.org/markovs-inequality) bounds \( \mathbb{P}(|S_n|/n > \varepsilon) \) by a summable quantity, and Borel–Cantelli I yields \( S_n/n \to 0 \) almost surely.<sup>[6](https://tsoo-math.github.io/ucl/borel.html)</sup>

**Modes of convergence.** Borel–Cantelli I also refines the relation between convergence in probability and almost sure convergence: \( X_n \to X \) in probability if and only if every subsequence has a further sub-subsequence converging almost surely, a result proved by choosing the sub-subsequence so that the deviation probabilities are summable.<sup>[11](https://web.stanford.edu/~montanar/TEACHING/Stat310B/NOTES/amir_notes.pdf)</sup>

**Metric Diophantine approximation.** The real numbers that are \( \psi \)-approximable form a limsup set, and Khintchine's theorem about the [Lebesgue measure](https://www.edgepedia.org/lebesgue-measure) of this set parallels the dichotomy in the Borel–Cantelli lemma: the measure equals 1 for \( \psi(q) = 1/(q \log q) \), while for \( \psi(q) = 1/(q(\log q)^{1+\varepsilon}) \), with \( \varepsilon \) any positive number, it equals 0. The convergence case of the proof is essentially Borel–Cantelli I, while the divergence case requires quasi-independence of the approximating sets rather than true independence.<sup>[2](https://arxiv.org/abs/0710.4264)</sup>

**Dynamical systems and Markov chains.** For a strictly stationary sequence \( (X_k) \) and Borel sets \( (A_k) \), conditions on the dependence structure ensure that the events \( \{X_k \in A_k\} \) satisfy the Borel–Cantelli property; for absolutely regular sequences, \( \mu(\limsup_n A_n) > 0 \) suffices. Versions of the lemma for systems with sufficiently fast decay of correlations, so-called dynamical Borel–Cantelli lemmas, extend the theory beyond independent events.<sup>[13](https://arxiv.org/abs/1904.01850)</sup><sup> • </sup><sup>[14](https://arxiv.org/abs/1201.5866)</sup>

## History

Borel's 1909 paper introduced what was called his Zero-One Law, also known as the Borel–Cantelli lemmas, alongside his strong law of large numbers and continued fraction theorem, as part of a theory of "denumerable probability" applied to decimal and continued fraction expansions of real numbers.<sup>[1](https://www.math.purdue.edu/~eremenko/Pdf/barone.pdf)</sup><sup> • </sup><sup>[15](https://www.mdpi.com/2673-9984/5/1/38)</sup>  Borel proved the result assuming total independence of the events; Cantelli observed that independence is not needed when the sum of probabilities converges, which is the first lemma in its modern form.<sup>[2](https://arxiv.org/abs/0710.4264)</sup> Francesco Paolo Cantelli (Palermo 1875 – Rome 1966) was an Italian mathematician and statistician whose approach to the foundations of probability differed markedly from Borel's, though his name is generally linked with Borel's in the lemma.<sup>[16](https://www.treccani.it/enciclopedia/cantelli_%28Enciclopedia-della-Matematica%29/)</sup><sup> • </sup><sup>[17](https://www.sciencedirect.com/science/article/pii/0315086088900493)</sup>  Historical analysis notes that Borel's original reasoning relied on countable independence without a careful justification, and that the "Cantelli modification" of the Zero-One Law repaired one of the flaws.<sup>[1](https://www.math.purdue.edu/~eremenko/Pdf/barone.pdf)</sup>

## References

1. [J. Barone and A. Novikoff, "A History of the Axiomatic Formulation of Probability from Borel to Kolmogorov: Part I"](https://www.math.purdue.edu/~eremenko/Pdf/barone.pdf)
2. [M. M. Dodson, "Diophantine approximation, Khintchine's theorem, torus geometry and Hausdorff dimension", arXiv:0710.4264](https://arxiv.org/abs/0710.4264)
3. [D. Miranda, "Independência", Probabilidade, Chapter 3, UFABC](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-03-independencia.html)
4. [A. Montanari, "Stat 310A lecture notes", Stanford University](https://web.stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)
5. ["Weak independence of events and the converse of the Borel–Cantelli Lemma", arXiv:2004.11324](https://arxiv.org/abs/2004.11324)
6. [T. Soo, "The strong law of large numbers and the Borel–Cantelli lemma", UCL](https://tsoo-math.github.io/ucl/borel.html)
7. ["Independence, product measures and the Borel–Cantelli lemmas", Thuztra Learn](https://learn.thuztra.com/reference/independence-and-the-borel-cantelli-lemmas)
8. ["The Strong Law of Large Numbers", lecture notes, University of Wrocław](https://www.math.uni.wroc.pl/~szekli/documents/m-CHAINS-21/B.pdf)
9. ["Borel–Cantelli Lemma", ScienceDirect Topics overview](https://www.sciencedirect.com/topics/mathematics/borel-cantelli-lemma)
10. ["Independence, Borel–Cantelli and Zero-One Laws", Alphabeta Math / Crash Statistics](https://crash-statistics.netlify.app/chapters/ch1-probability/ch4-independence)
11. [A. Montanari, "Stat 310B lecture notes", Stanford University](https://web.stanford.edu/~montanar/TEACHING/Stat310B/NOTES/amir_notes.pdf)
12. [T. K. Chandra, *The Borel–Cantelli Lemma*, Springer (2012)](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MV/Chandra%20T.K.%20The%20Borel-Cantelli%20lemma%20%28Springer%2C%202012%29%28ISBN%209788132206767%29%28O%29%28114s%29_MV_.pdf)
13. [J. Dedecker, F. Merlevède, E. Rio, "Criteria for Borel–Cantelli lemmas with applications to Markov chains and dynamical systems", arXiv:1904.01850](https://arxiv.org/abs/1904.01850)
14. [N. Luzia, "A Borel–Cantelli lemma and its applications", arXiv:1201.5866](https://arxiv.org/abs/1201.5866)
15. ["Borel and the Emergence of Probability on the Mathematical Scene in France", MDPI](https://www.mdpi.com/2673-9984/5/1/38)
16. ["Cantelli", Enciclopedia della Matematica, Treccani (2013)](https://www.treccani.it/enciclopedia/cantelli_%28Enciclopedia-della-Matematica%29/)
17. ["Un 'Probabilista neoclassico': Francesco Paolo Cantelli", ScienceDirect](https://www.sciencedirect.com/science/article/pii/0315086088900493)]

---
*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Convergence of random variables › Borel–Cantelli lemmas*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license