# Kolmogorov's three-series theorem

**Kolmogorov's three-series theorem** is a necessary and sufficient condition for the almost sure convergence (convergence with probability one) of a series of independent random variables, expressed through the convergence of three deterministic series built from tail probabilities, truncated means, and truncated variances. It is due to [Andrey Kolmogorov](https://www.edgepedia.org/andrey-kolmogorov)'s 1925 work in probability theory, and the name comes from the three conditions, which relate to tail probabilities, truncated means, and truncated variances.<sup>[1](https://mathshistory.st-andrews.ac.uk/Strick/kolmogorov.pdf)</sup>

| Key fact | Detail |
|---|---|
| Statement | For independent random variables \( X_1, X_2, \ldots \) and a constant \( C > 0 \), \( \sum_i X_i \) converges almost surely if and only if \( \sum_i P(|X_i| > C) \), \( \sum_i E[X_i \cdot 1_{\{|X_i| \le C\}}] \), and \( \sum_i \mathrm{Var}(X_i \cdot 1_{\{|X_i| \le C\}}) \) all converge<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> |
| Choice of cutoff | Convergence of the three series for one positive \( c \) implies almost sure convergence, and almost sure convergence implies the three series converge for all \( c \)<sup>[3](https://web.stanford.edu/~zyin/Theorems.pdf)</sup> |
| Precursor | Kolmogorov's two-series theorem gives a sufficient condition: if \( \sum_i E[X_i] \) and \( \sum_i \mathrm{Var}(X_i) \) converge, then \( \sum_i X_i \) converges almost surely<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> |
| Key tool | Kolmogorov's inequality, \( P(\max_{1 \le k \le n} |S_k| \ge l) \le \mathrm{Var}(S_n)/l^2 \) for independent zero-mean variables, controls the maximum of partial sums<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> |
| Random-signs corollary | For \( \varepsilon_k = \pm 1 \) with probability 1/2 each and bounded \( a_k \), the series \( \sum \varepsilon_k a_k \) converges with probability 1 if and only if \( \sum a_k^2 < \infty \)<sup>[4](https://nzdr.ru/data/media/biblio/kolxoz/M/MV/Skorokhod%20A.%2C%20Prokhorov%20I.%20Basic%20Principles%20and%20Applications%20of%20Probability%20Theory%20%28ISBN%203540546863%29%28Springer%2C%202005%29%28276s%29.pdf)</sup> |
| Extensions | Versions exist for independent products on Lie groups (Ming Liao, 2014) and for independent random variables under sub-linear expectations (Xu and Zhang, 2019)<sup>[5](https://arxiv.org/abs/1412.7845)</sup><sup> • </sup><sup>[6](https://actamath.cjoe.ac.cn/Jwk_sxxb_en/EN/10.1007/s10114-018-7508-9)</sup> |

## Statement

Let \( X_1, X_2, \ldots \) be independent random variables and fix a constant \( C > 0 \). The theorem states that \( \sum_i X_i \) converges almost surely if and only if the following three series all converge:<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup>

1. \( \sum_i P(|X_i| > C) \), the series of tail probabilities;
2. \( \sum_i E[X_i \cdot 1_{\{|X_i| \le C\}}] \), the series of truncated means;
3. \( \sum_i \mathrm{Var}(X_i \cdot 1_{\{|X_i| \le C\}}) \), the series of truncated variances.

Here \( 1_{\{|X_i| \le C\}} \) is the indicator of the event that \( |X_i| \) is at most \( C \), so \( X_i \cdot 1_{\{|X_i| \le C\}} \) is the variable with its large values cut off, or truncated, at level \( C \). The truncated variables are bounded, so their means and variances are finite term by term; the question the theorem answers is whether the sums of these quantities over all \( i \) are finite.

The cutoff level is remarkably flexible. Different values of \( C \) produce different truncated series, but they converge simultaneously: if the three series converge for some positive \( c \), then \( \sum X_n \) converges almost surely, and conversely almost sure convergence forces the three series to converge for every \( c \).<sup>[3](https://web.stanford.edu/~zyin/Theorems.pdf)</sup> This means a proof can choose whichever cutoff makes the analysis easiest.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup>

## Relation to the two-series theorem

The three-series theorem refines an earlier sufficient condition. Kolmogorov's two-series theorem says that if \( X_1, X_2, \ldots \) are independent and both \( \sum_i E[X_i] \) and \( \sum_i \mathrm{Var}(X_i) \) converge, then \( \sum_i X_i \) converges almost surely. In the common case of zero-mean variables this reduces to requiring \( \sum \mathrm{Var}(X_n) < \infty \).<sup>[3](https://web.stanford.edu/~zyin/Theorems.pdf)</sup> The two-series theorem gives only a sufficient condition; the three-series theorem supplies the matching necessary condition, completing the characterization.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup>

## Idea of the proof

**Sufficiency by truncation.** The forward direction of the proof sets \( Y_i = X_i \cdot 1_{\{|X_i| \le C\}} \). If the two-series theorem applies to the \( Y_i \), and if \( \sum_{i=1}^{\infty} P(X_i \neq Y_i) = \sum_{i=1}^{\infty} P(|X_i| > C) \) is finite, then the [Borel–Cantelli lemma](https://www.edgepedia.org/borel-cantelli-lemma) implies that \( X_i = Y_i \) for all but finitely many \( i \) with probability 1. The series \( \sum_i X_i \) then converges almost surely because \( \sum_i Y_i \) does.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> In the formulation of Skorokhod and Prokhorov, if the tail-probability series converges, then \( \sum \xi_k 1_{\{|\xi_k| > c\}} \) has only finitely many nonzero terms with probability 1, again by Borel–Cantelli, and the truncated part converges by the two-series result applied to centered bounded variables.<sup>[4](https://nzdr.ru/data/media/biblio/kolxoz/M/MV/Skorokhod%20A.%2C%20Prokhorov%20I.%20Basic%20Principles%20and%20Applications%20of%20Probability%20Theory%20%28ISBN%203540546863%29%28Springer%2C%202005%29%28276s%29.pdf)</sup>

**Kolmogorov's inequality.** The maximal inequality behind the two-series step states that for independent variables with zero mean and finite second moments,

\[ P\left( \max_{1 \le k \le n} |S_k| \ge l \right) \le \frac{1}{l^2} E\left[ (S_n)^2 \right], \]

where \( S_k = X_1 + \cdots + X_k \). Its proof is a stopping-time argument: one splits the event that some partial sum reaches \( l \) according to the first index at which this happens, and uses independence to show the cross term \( E[S_k (S_n - S_k) 1_{E_k}] \) vanishes.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> Applying the inequality to blocks \( X_{m+1}, \ldots, X_n \) shows that the partial sums are Cauchy with probability tending to 1 whenever the total variance is finite, which is how almost sure convergence is obtained.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup>

**Necessity.** The converse direction, showing that almost sure convergence forces all three series to converge, is the part the three-series theorem adds over the two-series theorem; standard treatments refer to the text for its details.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup>

## Applications

**Random signs.** A classical example asks what happens when the signs of a deterministic series are chosen by coin flips. From calculus, \( \sum_{n=1}^{\infty} 1/n^s \) converges if and only if \( s > 1 \), while the alternating series \( \sum (-1)^n / n^s \) converges for \( s > 0 \). If \( Y_1, Y_2, \ldots \) are independent with \( P(Y_n = +1) = P(Y_n = -1) = 1/2 \), the two-series theorem shows \( \sum Y_n / n^s \) converges almost surely for \( s > 1/2 \), since \( \mathrm{Var}(Y_n/n^s) = 1/n^{2s} \) and \( \sum 1/n^{2s} \) converges when \( 2s > 1 \). The three-series theorem then shows the series is almost surely divergent for \( s \le 1/2 \), so random cancellation pushes the convergence threshold from \( s > 1 \) down to \( s > 1/2 \) but no further.<sup>[2](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)</sup> More generally, for non-random \( a_n \) with \( \sum a_n^2 < \infty \) and independent Bernoulli variables of parameter \( p = 1/2 \), the series \( \sum (-1)^{B_n} a_n \) converges with probability one, even when \( \sum |a_n| = \infty \).<sup>[7](https://stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup>

**Random power series.** The theorem controls where random series converge on the boundary of their disk of convergence. For \( P(z) = \sum_{n \ge 1} \varepsilon_n a_n z^n \) with deterministic \( a_n \) and random signs \( \varepsilon_n \), Kolmogorov's three-series theorem implies that if \( \sum_n |a_n|^2 = \infty \), then \( P(z) \) almost surely diverges at almost every \( z \) with \( |z| = 1 \).<sup>[8](https://arxiv.org/abs/2509.02729)</sup> A 2025 paper by Marcus Michelen and Mehtaab Sawhney builds on this boundary to settle a 1961 question of [Paul Erdős](https://www.edgepedia.org/paul-erdos) about the size of the set of convergent points.<sup>[8](https://arxiv.org/abs/2509.02729)</sup>

**Connection to the strong law.** Convergence criteria for random series feed directly into proofs of the strong law of large numbers. In one route, for integrable i.i.d. variables one truncates at \( X_k 1_{\{|X_k| \le k\}} \), shows \( \sum_k k^{-2} \mathrm{Var}(X_k) \le 2 E|X_1| \), applies the series convergence theorem to \( Y_k = k^{-1}(X_k - E X_k) \), and converts the convergent series into the desired average limit using Kronecker's lemma.<sup>[7](https://stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)</sup>

## Extensions

The theorem has been carried into settings beyond sums of real random variables. In 2014, the [Auburn University](https://www.edgepedia.org/auburn-university) mathematician Ming Liao generalized Kolmogorov's three-series theorem to Lie groups, deriving criteria that are both necessary and sufficient for independent products on such groups to converge, and applied the result to independent random matrices.<sup>[5](https://arxiv.org/abs/1412.7845)</sup> Jiapan Xu and Lixin Zhang established a three-series theorem for independent random variables under sub-linear expectations, a framework in which the expectation and its related capacity are not additive, and used it to derive Marcinkiewicz's strong law of large numbers in that setting; the paper appeared in Acta Mathematica Sinica in 2019.<sup>[6](https://actamath.cjoe.ac.cn/Jwk_sxxb_en/EN/10.1007/s10114-018-7508-9)</sup> Work on pairwise negatively quadrant dependent sequences has also produced generalized Kolmogorov-type three-series results.<sup>[9](https://arxiv.org/abs/1303.4265)</sup>

## References

1. [Heinz Klaus Strick, "Andrei Kolmogorov" (biography), MacTutor History of Mathematics archive](https://mathshistory.st-andrews.ac.uk/Strick/kolmogorov.pdf)
2. [Probability: Limit Theorems I, lecture notes (Kolmogorov's inequality, two-series and three-series theorems)](https://homepages.math.uic.edu/~itobasco/courses/NYU/LimitThms1.pdf)
3. [Convergence of Random Series, theorem summary notes, Stanford](https://web.stanford.edu/~zyin/Theorems.pdf)
4. [A. V. Skorokhod, I. I. Prokhorov, Basic Principles and Applications of Probability Theory, Springer (2005), Section 3.2](https://nzdr.ru/data/media/biblio/kolxoz/M/MV/Skorokhod%20A.%2C%20Prokhorov%20I.%20Basic%20Principles%20and%20Applications%20of%20Probability%20Theory%20%28ISBN%203540546863%29%28Springer%2C%202005%29%28276s%29.pdf)
5. [Ming Liao (2014), "A three-series theorem on Lie groups", arXiv:1412.7845](https://arxiv.org/abs/1412.7845)
6. [Jia Pan Xu, Li Xin Zhang, "Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications", Acta Mathematica Sinica 35(2): 172-184 (2019)](https://actamath.cjoe.ac.cn/Jwk_sxxb_en/EN/10.1007/s10114-018-7508-9)
7. [Stat 310A lecture notes, Stanford (random signs, Kronecker's lemma, strong law)](https://stanford.edu/~montanar/TEACHING/Stat310A/lnotes.pdf)
8. [Marcus Michelen, Mehtaab Sawhney (2025), "Convergent points for random power series on the unit circle", arXiv:2509.02729](https://arxiv.org/abs/2509.02729)
9. ["The improvement about convergence properties for pairwise NQD random sequences", arXiv:1303.4265](https://arxiv.org/abs/1303.4265)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Probability theory › Random variables › Convergence of random variables › Strong law of large numbers*

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