# Kolmogorov's zero–one law

**Kolmogorov's zero–one law** is a theorem of probability theory stating that every tail event of a sequence of independent random variables, meaning an event whose occurrence is determined only by arbitrarily distant terms of the sequence, has probability either 0 or 1.<sup>[1](https://encyclopediaofmath.org/wiki/Zero-one_law)</sup> The law is named for Andrei Nikolaevich Kolmogorov (1903–1987), the Russian mathematician whose 1933 monograph *Grundbegriffe der Wahrscheinlichkeitsrechnung* laid the measure-theoretic foundations of modern probability.<sup>[2](https://www.sciencedirect.com/science/chapter/edited-volume/pii/B978044450871350156X)</sup>

| Key fact | Detail |
|---|---|
| Statement | If \( X_1, X_2, \ldots \) are independent random variables and \( A \) lies in the tail σ-algebra, then \( \mathbb{P}(A) \in \{0, 1\} \)<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup> |
| Tail σ-algebra | \( \mathcal{T} = \bigcap_{n=1}^{\infty} \sigma(X_n, X_{n+1}, \ldots) \), the events unaffected by any finite prefix of the sequence<sup>[4](https://androma.org/theorems/512)</sup> |
| Corollary | Any \( \mathcal{T} \)-measurable real random variable is almost surely constant<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup> |
| Proof idea | A tail event is independent of every finite initial segment, hence of the σ-algebra generated by all of them, and therefore of itself; so \( \mathbb{P}(A) = \mathbb{P}(A)^2 \)<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup> |
| Typical tail events | Convergence of \( X_n \), the value of \( \limsup_n X_n \), convergence of a random series, existence of an infinite cluster in percolation<sup>[5](https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/754f10a33d3e883d60c5db6de4fa24fc_MIT18_175S14_Lecture11.pdf)</sup><sup> • </sup><sup>[6](https://math.uchicago.edu/~may/REU2017/REUPapers/McNamara.pdf)</sup> |
| Scope | Requires independence; without it tail events can have intermediate probabilities, and the question connects to phase transitions<sup>[7](https://encyclopediaofmath.org/wiki/Tail_sigma-field)</sup> |

## Tail events and the tail σ-algebra

For a sequence of random variables \( X_1, X_2, \ldots \) on a probability space \( (\Omega, \mathcal{F}, \mathbb{P}) \), write \( \mathcal{F}_n = \sigma(X_n, X_{n+1}, \ldots) \) for the σ-algebra generated by all terms from the \( n \)-th onward. The tail σ-algebra is the intersection \( \mathcal{T} = \bigcap_{n=1}^{\infty} \mathcal{F}_n \); its elements are called tail events.<sup>[4](https://androma.org/theorems/512)</sup> Intuitively, \( \mathcal{T} \) contains the information observable by looking only arbitrarily far into the future: membership in a tail event does not change when finitely many of the \( X_n \) are altered.<sup>[5](https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/754f10a33d3e883d60c5db6de4fa24fc_MIT18_175S14_Lecture11.pdf)</sup>

Standard examples make the idea concrete. If the sequence is real-valued, the event that \( X_n \) converges to a limit is a tail event, since convergence depends only on the Cauchy behavior of terms with large indices.<sup>[5](https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/754f10a33d3e883d60c5db6de4fa24fc_MIT18_175S14_Lecture11.pdf)</sup> The events \( \{\limsup_n X_n = \infty\} \) and \( \{\limsup_n X_n \le 1\} \) are tail events, as is the event that the partial sums \( S_n = X_1 + \cdots + X_n \), suitably normalized, converge.<sup>[8](https://www2.stat.duke.edu/courses/Spring09/sta205/lec/indep.pdf)</sup> For a monkey typing random characters forever, the event of producing infinitely many copies of the Collected Works of Shakespeare is a tail event, because no finite prefix of the typing can affect it.<sup>[9](https://ma.huji.ac.il/~razk/Teaching/LectureNotes/LectureNotesProbability.pdf)</sup>

## Statement and proof

**The theorem.** If \( X_1, X_2, \ldots \) are independent random variables and \( A \in \mathcal{T} \), then \( \mathbb{P}(A) = 0 \) or \( \mathbb{P}(A) = 1 \).<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup> Equivalently, the tail σ-algebra is trivial: it contains only sets of probability measure 0 or 1.<sup>[10](https://nzdr.ru/data/media/biblio/kolxoz/M/MV/Gut%20A.%20Probability.%20A%20graduate%20course%20%282ed.%2C%20Springer%2C%202013%29%28ISBN%209781461447078%29%28O%29%28619s%29_MV_.pdf)</sup>

The classical proof rests on a single observation. Because \( A \in \mathcal{T} \), it lies in \( \sigma(X_{n+1}, X_{n+2}, \ldots) \) for every \( n \), and this σ-algebra is independent of \( \sigma(X_1, \ldots, X_n) \). Hence \( A \) is independent of every finite initial segment of the sequence, and by a limiting argument (via the monotone class or π-system criterion) it is independent of the σ-algebra generated by the whole sequence. Since \( \mathcal{T} \) itself sits inside that generated σ-algebra, \( A \) is independent of itself. Then

\[ \mathbb{P}(A) = \mathbb{P}(A \cap A) = \mathbb{P}(A) \cdot \mathbb{P}(A), \]

so \( \mathbb{P}(A) \) satisfies \( p = p^2 \), giving \( p \in \{0, 1\} \).<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup><sup> • </sup><sup>[8](https://www2.stat.duke.edu/courses/Spring09/sta205/lec/indep.pdf)</sup> The same argument shows that any \( \mathcal{T} \)-measurable random variable is almost surely constant: its level sets are tail events, so the variable cannot take two separated values with positive probability.<sup>[3](https://www.math.utah.edu/~davar/probtheory.pdf)</sup> In particular, for an independent sequence the quantities \( \limsup_n X_n \) and \( \liminf_n X_n \) are almost surely constant.<sup>[11](https://mathweb.ucsd.edu/~bdriver/280_06-07/Lecture_Notes/N7.pdf)</sup>

A martingale proof gives the same conclusion: for a tail event \( A \), the Doob martingale \( \mathbb{E}(\mathbf{1}_A \mid \mathcal{F}_n) \) equals the constant \( \mathbb{P}(A) \) for every \( n \), yet converges almost surely to \( \mathbf{1}_A \), forcing \( \mathbb{P}(A) = \mathbf{1}_A \).<sup>[12](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/17%3A_Martingales/17.05%3A_Convergence)</sup>

## What the law does and does not tell you

The law is a dichotomy result, not a computation tool. Knowing that a tail event has probability 0 or 1 does not identify which value holds; that usually requires separate analysis, such as [Kolmogorov's three-series theorem](https://www.edgepedia.org/kolmogorovs-three-series-theorem) for convergence of random series.<sup>[13](https://youyuanwu.github.io/misc-docs/math/11-probability/09-zero-one-laws.html)</sup> In the three-series theorem, the zero-one law supplies the first step: the event that \( \sum_n X_n \) converges is a tail event, so its probability is \( p \in \{0, 1\} \), and the remaining work is to show \( p = 1 \) when the three series conditions hold and \( p = 0 \) when any one fails.<sup>[5](https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/754f10a33d3e883d60c5db6de4fa24fc_MIT18_175S14_Lecture11.pdf)</sup> Similarly, for a random series \( \sum_n \omega_n a_n \) with independent signs, no finite number of sign switches can change convergence, so the convergence event lies in the tail field and has probability 0 or 1.<sup>[6](https://math.uchicago.edu/~may/REU2017/REUPapers/McNamara.pdf)</sup>

**Percolation.** In bond percolation on the square lattice \( \mathbb{Z}^2 \), each edge of the coordinate plane is retained independently with probability \( p \). The existence of an infinite connected component of open edges is a tail event, since ignoring any finite set of edges cannot create or destroy an infinite cluster. The zero-one law therefore implies \( \mathbb{P}_p(C_\infty) \in \{0, 1\} \) for every \( p \); the sharper result, beyond the law itself, is that the critical parameter is \( p_c = 1/2 \), with an infinite cluster occurring with probability one when \( p \ge 1/2 \) and probability zero when \( p < 1/2 \).<sup>[6](https://math.uchicago.edu/~may/REU2017/REUPapers/McNamara.pdf)</sup>

**Random power series.** For a random power series \( f(z, \omega) = \sum_{i=0}^{\infty} X_i(\omega) z^i \) built from independent coefficients, the radius of convergence is a tail-measurable quantity, so it is constant across almost all outcomes \( \omega \).<sup>[6](https://math.uchicago.edu/~may/REU2017/REUPapers/McNamara.pdf)</sup> In the classical setting of independent random variables uniformly distributed on \( (0,1) \), B. Jessen proved that any tail event has probability 0 or 1, and the event that such a series cannot be extended across the boundary of its disc of convergence has probability 1.<sup>[1](https://encyclopediaofmath.org/wiki/Zero-one_law)</sup>

## Related zero-one laws and extensions

The zero-one law belongs to a family of results. For independent events \( A_n \), the Borel–Cantelli lemmas already give a zero-one dichotomy for the event that infinitely many \( A_n \) occur: its probability is 0 when \( \sum_n \mathbb{P}(A_n) < \infty \) and 1 when the \( A_n \) are independent and \( \sum_n \mathbb{P}(A_n) = \infty \).<sup>[8](https://www2.stat.duke.edu/courses/Spring09/sta205/lec/indep.pdf)</sup> For independent and identically distributed sequences, the Hewitt–Savage zero-one law reaches further: the probability of any event invariant under permutations of finitely many terms of the sequence is also 0 or 1.<sup>[1](https://encyclopediaofmath.org/wiki/Zero-one_law)</sup> The two laws decide different events; the exchangeable σ-algebra can be nontrivial for merely independent, non-identically-distributed sequences.<sup>[13](https://youyuanwu.github.io/misc-docs/math/11-probability/09-zero-one-laws.html)</sup>

Independence is essential. For dependent sequences the tail σ-algebra can carry events of intermediate probability, and tail triviality then holds only under sufficiently weak dependence; for random fields the question connects to phase transition, and tail triviality holds for extremal Gibbs measures.<sup>[7](https://encyclopediaofmath.org/wiki/Tail_sigma-field)</sup> Analogues have been developed in other settings: a version of the law for homogeneous and nonhomogeneous Markov chains states that for large \( n \), with probability near one, the chain sits in states \( i \) for which \( P(A \mid Z_n = i) \) is near 0 or near 1 for any tail event \( A \),<sup>[14](https://arxiv.org/abs/2011.04063)</sup> and categorical-probability frameworks yield general versions of the Kolmogorov and Hewitt–Savage laws that specialize to the standard theorems on standard Borel spaces.<sup>[15](https://arxiv.org/abs/1912.02769)</sup>

## Historical setting

The law appears in the tradition founded by Kolmogorov's *Grundbegriffe der Wahrscheinlichkeitsrechnung* (1933), the short book that expressed probability in terms of set and measure theory and became the standard foundation of the subject; its appendix includes the zero-or-one law.<sup>[2](https://www.sciencedirect.com/science/chapter/edited-volume/pii/B978044450871350156X)</sup><sup> • </sup><sup>[16](https://www.mathematik.com/Kolmogorov/)</sup> Earlier work by [Émile Borel](https://www.edgepedia.org/emile-borel) had already produced zero-one phenomena for independent events, in the results now known as the Borel–Cantelli lemmas, and Kolmogorov's theorem generalized this dichotomy to arbitrary tail events of independent sequences.<sup>[17](https://philpapers.org/rec/BARAHO-18)</sup>

## References

1. [Zero-one law, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Zero-one_law)
2. [Jan von Plato, "A.N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (1933)", Landmark Writings in Western Mathematics 1640–1940, Elsevier (2005)](https://www.sciencedirect.com/science/chapter/edited-volume/pii/B978044450871350156X)
3. [Probability theory lecture notes, University of Utah (Theorem 5.15)](https://www.math.utah.edu/~davar/probtheory.pdf)
4. [Kolmogorov Zero One Law — Statement & Proof, Androma](https://androma.org/theorems/512)
5. [Scott Sheffield, 18.175 Theory of Probability, Lecture 11, MIT OpenCourseWare](https://ocw.mit.edu/courses/18-175-theory-of-probability-spring-2014/754f10a33d3e883d60c5db6de4fa24fc_MIT18_175S14_Lecture11.pdf)
6. [Declan McNamara, "Kolmogorov's Zero-One Law with Applications", University of Chicago REU (2017)](https://math.uchicago.edu/~may/REU2017/REUPapers/McNamara.pdf)
7. [Tail sigma-field, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Tail_sigma-field)
8. [Lecture notes on independence, STA 205, Duke University](https://www2.stat.duke.edu/courses/Spring09/sta205/lec/indep.pdf)
9. [Raz Kupferman, Lectures on Probability Theory, Hebrew University](https://ma.huji.ac.il/~razk/Teaching/LectureNotes/LectureNotesProbability.pdf)
10. [Allan Gut, Probability: A Graduate Course, 2nd ed., Springer (2013), Chapter 5](https://nzdr.ru/data/media/biblio/kolxoz/M/MV/Gut%20A.%20Probability.%20A%20graduate%20course%20%282ed.%2C%20Springer%2C%202013%29%28ISBN%209781461447078%29%28O%29%28619s%29_MV_.pdf)
11. [Lecture notes on stochastic processes, UC San Diego](https://mathweb.ucsd.edu/~bdriver/280_06-07/Lecture_Notes/N7.pdf)
12. [17.5 Convergence, Statistics LibreTexts (Siegrist)](https://stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/17%3A_Martingales/17.05%3A_Convergence)
13. [Zero-One Laws, notes by Youyuan Wu](https://youyuanwu.github.io/misc-docs/math/11-probability/09-zero-one-laws.html)
14. [A Zero-One Law for Markov Chains, arXiv:2011.04063](https://arxiv.org/abs/2011.04063)
15. [Infinite products and zero-one laws in categorical probability, arXiv:1912.02769](https://arxiv.org/abs/1912.02769)
16. [Kolmogorov: Foundations of the Theory of Probability](https://www.mathematik.com/Kolmogorov/)
17. [Jack Barone & Albert Novikoff, "A History of the Axiomatic Formulation of Probability from Borel to Kolmogorov: Part I", Archive for History of Exact Sciences 18 (1978)](https://philpapers.org/rec/BARAHO-18)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Exchangeability, independence, and Gaussian structure › Independent and identically distributed sequences*

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