# Hewitt–Savage zero–one law

The **Hewitt–Savage zero–one law** is a theorem of probability theory stating that if \( X_1, X_2, \ldots \) are independent and identically distributed (i.i.d.) random variables, then every event whose occurrence is invariant under any finite permutation of the variables has probability either 0 or 1.<sup>[1](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)</sup> It was proved by [Edwin Hewitt](https://www.edgepedia.org/edwin-hewitt) and Leonard J. Savage in their 1955 paper *Symmetric measures on Cartesian products*, published in the Transactions of the American Mathematical Society.<sup>[2](https://bishtref.com/articles/10.1090/s0002-9947-1955-0076206-8)</sup>

| Key fact | Detail |
|---|---|
| Statement | If \( X_1, X_2, \ldots \) are i.i.d. and \( A \) is exchangeable, then \( P(A) \in \{0,1\} \)<sup>[1](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)</sup> |
| Exchangeable events | Events invariant under every finite permutation, that is, every bijection of \( \mathbb{N} \) fixing all but finitely many indices; they form the exchangeable σ-algebra \( \mathcal{E} \)<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup> |
| Origin | Edwin Hewitt and Leonard J. Savage, *Symmetric measures on Cartesian products*, Trans. Amer. Math. Soc. 80(2), pp. 470–501 (1955)<sup>[2](https://bishtref.com/articles/10.1090/s0002-9947-1955-0076206-8)</sup> |
| Proof mechanism | Approximate \( A \) by an event depending on the first \( n \) coordinates, swap the blocks \( \{1,\ldots,n\} \) and \( \{n+1,\ldots,2n\} \), and conclude \( P(A) = P(A)^2 \)<sup>[4](https://math.mit.edu/~sheffield/175/Lecture23.pdf)</sup> |
| Relation to Kolmogorov's law | The tail σ-algebra is contained in \( \mathcal{E} \), so Hewitt–Savage is stronger; events such as \( \{S_n = 0 \text{ infinitely often}\} \) for a random walk are exchangeable but not tail events<sup>[5](https://etheses.dur.ac.uk/12498/1/C.H.LO_eThesis_osRWp.pdf?DDD21+=)</sup> |
| Signature application | For a random walk on \( \mathbb{R} \), exactly one of four limiting behaviors occurs almost surely<sup>[4](https://math.mit.edu/~sheffield/175/Lecture23.pdf)</sup> |
| Modern extensions | Information-theoretic proofs (O'Connell, 2000), positional-symmetry generalizations (2024), and a non-commutative version for quantum stochastic processes<sup>[6](https://www.aub.edu.lb/fas/math/Documents/zero-one-laws-report.pdf)</sup><sup> • </sup><sup>[7](https://arxiv.org/abs/2406.14902)</sup><sup> • </sup><sup>[8](https://arxiv.org/abs/2607.20342)</sup> |

## Exchangeable events

A bijection \( \pi \colon \mathbb{N} \to \mathbb{N} \) is a **finite permutation** if there is some \( N_0 \) such that \( \pi(n) = n \) for all \( n > N_0 \).<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup> Given a sequence \( \xi = (\xi_1, \xi_2, \ldots) \), applying \( \pi \) produces the rearranged sequence \( T_\pi \xi = (\xi_{\pi(1)}, \xi_{\pi(2)}, \ldots) \). An event \( A \) in the σ-algebra generated by the sequence is **symmetric** (or exchangeable, or permutable) if \( \{\xi \in B\} = \{T_\pi \xi \in B\} \) for every finite permutation \( \pi \). The symmetric events form a σ-algebra \( \mathcal{E} \), called the exchangeable σ-algebra.<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup>

In words, an exchangeable event is one whose occurrence is not affected by permuting any finite subset of the terms of the sequence.<sup>[6](https://www.aub.edu.lb/fas/math/Documents/zero-one-laws-report.pdf)</sup> The theorem then states that for an i.i.d. sequence taking values in a measurable space \( (S, \mathcal{S}) \), the exchangeable σ-algebra is trivial: if \( A \in \mathcal{E} \), then \( P(A) \in \{0, 1\} \).<sup>[5](https://etheses.dur.ac.uk/12498/1/C.H.LO_eThesis_osRWp.pdf?DDD21+=)</sup>

## Proof idea

The standard proof shows that \( A \) is independent of itself, so that \( P(A) = P(A \cap A) = P(A)^2 \), which forces \( P(A) \) to be 0 or 1.<sup>[4](https://math.mit.edu/~sheffield/175/Lecture23.pdf)</sup> The self-independence is not literal and must be arranged through approximation:

1. Every event in the product σ-algebra can be approximated by cylinder events: for any \( I \) there are events \( I_n \in \sigma(X_1, \ldots, X_n) \) with \( P(I_n \,\Delta\, I) \to 0 \), where \( \Delta \) denotes symmetric difference.<sup>[1](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)</sup>
2. Fix \( n \) and let \( \pi \) be the permutation that swaps the blocks \( \{1, \ldots, n\} \) and \( \{n+1, \ldots, 2n\} \) and fixes the remaining indices. Because the sequence is i.i.d., its law is invariant under \( \pi \); because \( A \) is symmetric, the permuted approximation \( C' \) of \( C \) satisfies \( P(A \,\Delta\, C') = P(A \,\Delta\, C) \) and \( P(C') = P(C) \).<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup>
3. \( C \) depends on the first \( n \) coordinates and \( C' \) on the next \( n \), so they are independent and \( P(C \cap C') = P(C)^2 \). Since \( P(A \,\Delta\, (C \cap C')) < 2\varepsilon \), letting \( \varepsilon \downarrow 0 \) gives \( P(A) = P(A)^2 \).<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup>

An alternative proof, due to N. O'Connell (2000), is information-theoretic. It uses the inequality \( H(V) \ge \sum_{i=1}^{n} I(V; W_i) \) for independent \( W_1, \ldots, W_n \), where \( H \) is entropy and \( I \) mutual information. Applied to the indicator of an exchangeable event, it shows the event is independent of every finite block of coordinates, hence of the whole sequence, hence of itself.<sup>[6](https://www.aub.edu.lb/fas/math/Documents/zero-one-laws-report.pdf)</sup>

## Relation to Kolmogorov's zero–one law

[Kolmogorov's zero–one law](https://www.edgepedia.org/kolmogorovs-zero-one-law) concerns **tail events**, whose occurrence is unaffected by changing finitely many terms of the sequence; for independent random variables, every tail event has probability 0 or 1.<sup>[6](https://www.aub.edu.lb/fas/math/Documents/zero-one-laws-report.pdf)</sup> The tail σ-algebra \( \mathcal{T} \) is contained in the exchangeable σ-algebra \( \mathcal{E} \), so the Hewitt–Savage law applies to strictly more events.<sup>[9](https://math.swansonsite.com/wp-content/uploads/2024/11/prob_notes.pdf)</sup>

The difference matters in practice. For a random walk \( S_n = X_1 + \cdots + X_n \) with i.i.d. increments, the event \( R = \{S_n = 0 \text{ infinitely often}\} \) is invariant under finite permutations of the increments and therefore exchangeable, but it is not a tail event, so Kolmogorov's law cannot be applied to it directly.<sup>[5](https://etheses.dur.ac.uk/12498/1/C.H.LO_eThesis_osRWp.pdf?DDD21+=)</sup> The same event \( \{S_n \in E \text{ infinitely often}\} \) for a set \( E \subset \mathbb{R}^d \) satisfies the Hewitt–Savage hypothesis but not that of the Kolmogorov version.<sup>[10](https://perso.ens-lyon.fr/thomas.budzinski/RW.pdf)</sup>

## Applications to random walks and Brownian motion

**Random walk dichotomy.** For a random walk \( (S_n) \) on \( \mathbb{R} \), exactly one of the following occurs almost surely: \( S_n = 0 \) for all \( n \); \( S_n \to \infty \); \( S_n \to -\infty \); or \( \liminf_n S_n = -\infty \) and \( \limsup_n S_n = \infty \).<sup>[4](https://math.mit.edu/~sheffield/175/Lecture23.pdf)</sup> The proof uses that \( \limsup_n S_n \) and \( \liminf_n S_n \) are exchangeable random variables, hence almost sure constants in \( [-\infty, \infty] \); if either were finite, the shifted walk \( S_n' = \xi_2 + \cdots + \xi_{n+1} \), which has the same law, would force \( \xi_1 = 0 \) almost surely, leaving only the degenerate case \( S_n = 0 \) for all \( n \).<sup>[3](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)</sup> The law also underlies the recurrence classification: by the Hewitt–Savage law, the probability that a walk visits a neighborhood of a point infinitely often is 0 or 1, so a recurrent value is one for which this probability is 1.<sup>[10](https://perso.ens-lyon.fr/thomas.budzinski/RW.pdf)</sup>

**Brownian motion.** Taking \( X_n = B_n - B_{n-1} \) for standard [Brownian motion](https://www.edgepedia.org/brownian-motion) \( (B_n) \), the law applies to events such as \( \{B_n > c\sqrt{n} \text{ infinitely often}\} \), whose probability is 0 or 1; Fatou's lemma rules out 0. This yields \( \limsup B_n / \sqrt{n} = +\infty \) almost surely, a statement about a nowhere-differentiable process in continuous time, obtained from a theorem about a countable sequence of random variables and finite permutations.<sup>[11](https://www.invariants.org.uk/wp-content/uploads/2023/02/InvariantMT22.pdf)</sup><sup> • </sup><sup>[16](https://www.stat.berkeley.edu/~aldous/205B/bmbook.pdf)</sup>

## Relation to de Finetti's theorem

The 1955 Hewitt–Savage paper extended de Finetti's representation theorem beyond binary random variables to more general state spaces under topological conditions: an exchangeable sequence can be represented as a mixture of i.i.d. sequences. The combined result is often called the de Finetti–Hewitt–Savage theorem.<sup>[12](https://arxiv.org/abs/2304.03927)</sup> The original de Finetti formulation covered exchangeable sequences of \( \{0,1\} \)-valued random variables; Hewitt and Savage obtained the same conclusion for more general state spaces under some topological conditions.<sup>[13](https://arxiv.org/abs/2008.08754)</sup>

A related result, often quoted in connection with de Finetti's theorem as the Hewitt–Savage lemma, asserts that the fixed-point σ-algebra of the permutation group coincides with the tail σ-algebra of the shift.<sup>[14](https://encyclopediaofmath.org/wiki/De_Finetti_theorem)</sup> In the exchangeable case, conditioning on the exchangeable σ-algebra makes the variables conditionally i.i.d.<sup>[15](https://web.stanford.edu/~zyin/Theorems.pdf)</sup>

## Generalizations

A 2024 line of work proves, with O'Connell's information-theoretic argument, that every event concerning a countably infinite family of i.i.d. random variables that has enough **positional symmetries** is deterministic: if for every finite \( J \) the event admits an injective symmetry \( \pi \) with \( \pi(J) \cap J = \emptyset \), its probability is 0 or 1. This result encompasses the Hewitt–Savage law, the zero–one law for shift-invariant events, and the ergodicity of the [Bernoulli process](https://www.edgepedia.org/bernoulli-process), and it also applies to infinite random graphs and simple renormalization processes.<sup>[7](https://arxiv.org/abs/2406.14902)</sup> The i.i.d. condition can be relaxed in this framework.<sup>[7](https://arxiv.org/abs/2406.14902)</sup>

Extensions have also reached beyond classical probability. A non-commutative Hewitt–Savage 0–1 law has been established for quantum stochastic processes, where tail triviality is equivalent to a factorization property of the distribution; in the commutative setting these notions collapse to the classical law.<sup>[8](https://arxiv.org/abs/2607.20342)</sup> Weighted generalizations of exchangeability, in which weight functions modify the individual distributions along the sequence, likewise carry extensions of the zero–one law.<sup>[12](https://arxiv.org/abs/2304.03927)</sup>

## References

1. [Probability Theory 1 Lecture Notes (Cornell, Jerison), Theorem 16.1 and Lemma 16.1](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)
2. [Edwin Hewitt, Leonard J. Savage (1955). Symmetric measures on Cartesian products. Transactions of the American Mathematical Society 80(2), 470–501.](https://bishtref.com/articles/10.1090/s0002-9947-1955-0076206-8)
3. [Medida Produto, Probabilidade (UFABC), Seção 8.8.1, Lei 0–1 de Hewitt–Savage](https://danielmiranda.prof.ufabc.edu.br/livros/prob/capitulo-08-medida-produto.html)
4. [18.175 Lecture 23: Random walks (MIT, Sheffield)](https://math.mit.edu/~sheffield/175/Lecture23.pdf)
5. [C. H. Lo, Durham E-Thesis, Theorem 6.3.1 (Hewitt–Savage zero-one law)](https://etheses.dur.ac.uk/12498/1/C.H.LO_eThesis_osRWp.pdf?DDD21+=)
6. [Information-theoretic proofs of zero-one laws (AUB), O'Connell's proof](https://www.aub.edu.lb/fas/math/Documents/zero-one-laws-report.pdf)
7. [Ayach, Khairallah, Manoukian, Mchaimech, Salha, Taati (2024). Zero-one laws for events with positional symmetries. arXiv:2406.14902](https://arxiv.org/abs/2406.14902)
8. [A Dynamical Approach to Non-Commutative de Finetti Theory. arXiv:2607.20342](https://arxiv.org/abs/2607.20342)
9. [Lecture Notes on Probability Theory, Theorem 11.5 (Hewitt–Savage 0-1 law)](https://math.swansonsite.com/wp-content/uploads/2024/11/prob_notes.pdf)
10. [Thomas Budzinski, Random walks notes, The Hewitt–Savage 0-1 law](https://perso.ens-lyon.fr/thomas.budzinski/RW.pdf)
11. [The Invariant, Cambridge Mathematical Tripos magazine (2022)](https://www.invariants.org.uk/wp-content/uploads/2023/02/InvariantMT22.pdf)
12. [De Finetti's theorem and related results for infinite weighted exchangeable sequences. arXiv:2304.03927](https://arxiv.org/abs/2304.03927)
13. [Irfan Alam (2020). Generalizing the de Finetti–Hewitt–Savage theorem. arXiv:2008.08754](https://arxiv.org/abs/2008.08754)
14. [De Finetti theorem, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/De_Finetti_theorem)
15. [Theorems cheat sheet (Stanford), item 37: Hewitt–Savage 0-1 law](https://web.stanford.edu/~zyin/Theorems.pdf)
16. [stat.berkeley.edu](https://www.stat.berkeley.edu/~aldous/205B/bmbook.pdf)

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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 › Exchangeability and de Finetti theory*

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