# Law of the iterated logarithm

The **law of the iterated logarithm** (LIL) is a limit theorem in probability theory that refines the strong law of large numbers: it identifies, almost surely, the exact envelope of the fluctuations of a random walk or [Brownian motion](https://www.edgepedia.org/brownian-motion), a boundary of order \( \sqrt{2 t \log \log t} \) that the process touches infinitely often but does not cross in the limit.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup><sup> • </sup><sup>[2](https://jovanaeducation.com/glossary/prob2-law-of-the-iterated-logarithm)</sup>

| Key fact | Detail |
|---|---|
| Statement (i.i.d. case) | For i.i.d. \( X_i \) with \( \mathsf{E}X = 0 \), \( \mathsf{E}X^2 = 1 \), and \( S_n = X_1 + \cdots + X_n \), \( \limsup_{n\to\infty} S_n / \sqrt{2 n \log \log n} = 1 \) almost surely<sup>[3](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup> |
| Brownian motion form | \( \limsup_{t\to\infty} B(t)/\sqrt{2 t \log \log t} = 1 \) a.s., and by time inversion \( \limsup_{t\downarrow 0} B(t)/\sqrt{2 t \log \log(1/t)} = 1 \) a.s.<sup>[3](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup> |
| First result | A. Ya. Khinchin, 1924, for Bernoulli trials (Fundamenta Mathematicae 6, pp. 9–20)<sup>[4](https://www.mdpi.com/2227-7390/13/5/867)</sup><sup> • </sup><sup>[5](https://www.sciencedirect.com/science/article/pii/S2590037424000451)</sup>  |
| General theorem | A. N. Kolmogorov, 1929, Mathematische Annalen 101, pp. 126–135, for independent non-identically distributed variables under a boundedness condition<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup> |
| i.i.d. theorem | Hartman and Wintner, 1941: finite nonzero variance \( b = \mathsf{E}X_1^2 \) gives \( c_n = (2 n b \ln \ln (n b))^{1/2} \)<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup> |
| Functional form | Volker Strassen, 1964, an invariance principle for the LIL in *Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*<sup>[6](https://portal.mardi4nfdi.de/wiki/An_invariance_principle_for_the_law_of_the_iterated_logarithm)</sup> |
| Statistical use | Iterated logarithm inequalities underlie confidence sequences and cumulative meta-analysis boundaries that control type I error over repeated looks at accumulating data<sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0167715299000607)</sup><sup> • </sup><sup>[8](https://www3.stat.sinica.edu.tw/statistica/j13n4/j13n411/j13n411.html)</sup>  |

## What the law says

While the strong law of large numbers gives conditions under which \( S_n / a_n \to 0 \) almost surely for a deterministic sequence \( a_n \), the law of the iterated logarithm concerns sequences \( \{c_n\} \) that separate two behaviors: with probability one, \( S_n \) exceeds members of the family \( \{c_n\} \) only finitely often for some choices and infinitely often for others.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup> In the classical normalization this is expressed as, for every \( \epsilon > 0 \),

\[ \mathsf{P}\{S_n > (1+\epsilon)\, c_n \text{ infinitely often}\} = 0, \qquad \mathsf{P}\{S_n > (1-\epsilon)\, c_n \text{ infinitely often}\} = 1. \]

The boundary is therefore sharp: any constant factor above 1 eventually excludes the walk, any factor below 1 is crossed infinitely often.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>

The iterated logarithm \( \log \log n \) grows extremely slowly, so the envelope \( \sqrt{2 n \log \log n} \) sits between the two standard limit theorems. The law of large numbers says \( S_n/n \to 0 \), and the central limit theorem describes fluctuations of order \( \sqrt{n} \) in distribution; the LIL gives the almost-sure extreme size of those \( \sqrt{n} \)-scale fluctuations, larger than \( \sqrt{n} \) by the slowly growing \( \sqrt{\log \log n} \) factor.<sup>[2](https://jovanaeducation.com/glossary/prob2-law-of-the-iterated-logarithm)</sup><sup> • </sup><sup>[9](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)</sup>  One reading is that the LIL identifies the scaling at which the almost-sure limit and the limit in probability differ: \( S_n/\sqrt{2 n \log \log n} \) tends to 0 in probability while its limsup (largest limit value a sequence repeatedly reaches) equals 1 almost surely.<sup>[9](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)</sup>

## History

**Khinchin and Kolmogorov.** The first theorem of general type on the law was obtained by Aleksandr Yakovlevich Khinchin (Хинчин), a Soviet mathematician and one of the founders of the Soviet school of probability, who proved it in 1924 for sums of independent variables taking two possible values (Bernoulli trials).<sup>[4](https://www.mdpi.com/2227-7390/13/5/867)</sup><sup> • </sup><sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>  Andrey Nikolaevich Kolmogorov (1903–1987), the author of the axiomatic foundations of probability published in 1933, extended the result in his 1929 paper *Über das Gesetz des iterierten Logarithmus* to independent random variables with zero means and finite variances that need not be identically distributed.<sup>[10](https://kam.mff.cuni.cz/~klazar/l10L_PRTE_20.pdf)</sup>Kolmogorov's theorem takes \( B_n = \sum_{k=1}^{n} \mathsf{E}X_k^2 \) and assumes \( B_n \to \infty \) together with a uniform asymptotic negligibility condition: there are constants \( M_n \) with \( |X_n| \le M_n \) and

\[ M_n = o\left( \left( \frac{B_n}{\ln \ln B_n} \right)^{1/2} \right). \]

Under these hypotheses the relations above hold with \( c_n = (2 B_n \ln \ln B_n)^{1/2} \).<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup> The condition requires each summand to be negligible compared with the norming factor, the same role it plays in the central limit theorem for triangular settings.<sup>[11](https://citeseerx.ist.psu.edu/document?doi=988528df1587305e47ced19111113f6be9c7795e&repid=rep1&type=pdf)</sup> The bound is tight in the sense that J. Marcinkiewicz and A. Zygmund showed in 1937 that under Kolmogorov's conditions the little-o cannot be weakened to big-O.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>

**Hartman and Wintner.** Philip Hartman and [Aurel Wintner](https://www.edgepedia.org/aurel-wintner), American mathematicians at [Johns Hopkins](https://www.edgepedia.org/johns-hopkins), proved in 1941 that for independent identically distributed variables with mean zero and \( 0 < \mathsf{E}X_1^2 < \infty \), the law holds with \( c_n = (2 n b \ln \ln (n b))^{1/2} \), where \( b = \mathsf{E}X_1^2 \).<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup> Their paper also characterized the law as a family of bounds of one order such that only finitely many partial sums exceed some members of the family while infinitely many exceed others.

**Feller and later generalizations.** [William Feller](https://www.edgepedia.org/william-feller) examined a generalization of Kolmogorov's law for sequences of independent bounded but non-identically distributed random variables, and further extensions cover dependent sequences, vectors, and random processes.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>

## Brownian motion and the functional form

For standard Brownian motion \( B(t) \), the law states

\[ \limsup_{t\to\infty} \frac{B(t)}{\sqrt{2 t \log \log t}} = 1 \quad \text{a.s.} \]

and, since \( \hat B(t) = t B(1/t) \) is again a Brownian motion, the same statement holds as \( t \downarrow 0 \) with \( \log \log(1/t) \) in place of \( \log \log t \), controlling the fine local roughness of the path near a point.<sup>[3](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)</sup><sup> • </sup><sup>[2](https://jovanaeducation.com/glossary/prob2-law-of-the-iterated-logarithm)</sup>  The liminf is \(-1\), so the path oscillates symmetrically between the two envelopes.<sup>[12](https://warwick.ac.uk/fac/sci/masdoc/people/studentpages/students2013/egginton/bm_lec_notes2.pdf)</sup> The growth is compatible with non-differentiability: Brownian motion oscillates at rate \( \sqrt{t \log \log(1/t)} \) near a point, infinitely faster than the linear rate differentiability would require.<sup>[13](https://github.com/SungchulLee/financial_math_book_writing/blob/main/docs/ch02/brownian_motion/brownian_motion.md)</sup>

In 1964 [Volker Strassen](https://www.edgepedia.org/volker-strassen), a German mathematician then working in probability theory, proved an invariance principle for the LIL, a functional form describing the set of limit points of the normalized partial-sum process; this functional law prefigures and generalizes the scalar statement.<sup>[6](https://portal.mardi4nfdi.de/wiki/An_invariance_principle_for_the_law_of_the_iterated_logarithm)</sup><sup> • </sup><sup>[4](https://www.mdpi.com/2227-7390/13/5/867)</sup>  Strassen's form of the law has since been extended to martingales, diffusion processes, empirical processes, Banach-space-valued walks, and, in recent work, to sub-linear expectation spaces and Markov processes.<sup>[6](https://portal.mardi4nfdi.de/wiki/An_invariance_principle_for_the_law_of_the_iterated_logarithm)</sup><sup> • </sup><sup>[14](https://arxiv.org/abs/2103.01390)</sup>

## What happens without finite variance

Finite variance is essential in the i.i.d. case. If the variables are i.i.d. with mean zero but infinite variance, the classical normalization fails in a strong way:

\[ \limsup_{n\to\infty} \frac{|S_n|}{(n \ln \ln n)^{1/2}} = \infty \quad \text{a.s.} \]<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>

so infinite variance guarantees that partial sums exceed the \( \sqrt{n \ln \ln n} \) envelope by an unbounded factor, though not generally the larger \( \sqrt{n \ln n \ln \ln n} \) envelope.<sup>[1](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)</sup>

## Statistical applications

**Confidence sequences.** Iterated logarithm inequalities were first used for statistical inference by D. A. Darling and [Herbert Robbins](https://www.edgepedia.org/herbert-robbins), who in 1967 built confidence sequences for the mean, median, and variance of i.i.d. observations, that is, confidence intervals valid at every sample size simultaneously.<sup>[7](https://www.sciencedirect.com/science/article/abs/pii/S0167715299000607)</sup><sup> • </sup><sup>[15](https://pmc.ncbi.nlm.nih.gov/articles/PMC335597/)</sup>  The LIL scaling gives time-uniform bounds whose width grows like \( \sqrt{\log \log n / n} \), tighter for most sample sizes than the \( \sqrt{\log n / n} \) width obtained from a naive union bound over all sample sizes.<sup>[16](https://proceedings.mlr.press/v108/dalalyan20a/dalalyan20a.pdf)</sup>

**Sequential monitoring.** The same boundary logic appears in cumulative meta-analysis, where an LIL-motivated penalty on the Z-value of a test statistic accounts for repeated inspections of accumulating data; simulations showed control of the overall type I error across a broad range of practical situations for up to 25 inspections.<sup>[8](https://www3.stat.sinica.edu.tw/statistica/j13n4/j13n411/j13n411.html)</sup> Nonasymptotic versions of the LIL, valid for every sample size and for general M-estimators rather than only the sample mean, have been applied to sequential tasks such as best-arm identification in multi-armed bandits with heavy-tailed rewards.<sup>[16](https://proceedings.mlr.press/v108/dalalyan20a/dalalyan20a.pdf)</sup>

## References

1. [Law of the iterated logarithm, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Law_of_the_iterated_logarithm)
2. [The law of the iterated logarithm, JOVANA Glossary](https://jovanaeducation.com/glossary/prob2-law-of-the-iterated-logarithm)
3. [Probability Theory 205B lecture notes, Lecture 23 (S. Chewi, UC Berkeley)](https://www.stat.berkeley.edu/users/aldous/205B/chewi_notes.pdf)
4. [The Heroic Age of Probability: Kolmogorov, Doob, Lévy, Khinchin and Feller, Mathematics 13(5):867 (2025)](https://www.mdpi.com/2227-7390/13/5/867)
5. [A general law of the iterated logarithm for non-additive probabilities, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S2590037424000451)
6. [An invariance principle for the law of the iterated logarithm, V. Strassen (1964), MaRDI portal](https://portal.mardi4nfdi.de/wiki/An_invariance_principle_for_the_law_of_the_iterated_logarithm)
7. [On Darling–Robbins type confidence sequences and sequential tests with power one, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0167715299000607)
8. [Applying the law of iterated logarithm to cumulative meta-analysis of a continuous endpoint, Statistica Sinica 13(4)](https://www3.stat.sinica.edu.tw/statistica/j13n4/j13n411/j13n411.html)
9. [Probability Theory 1 Lecture Notes (D. Jerison, Cornell)](https://pi.math.cornell.edu/~jerison/math6710/notes.pdf)
10. [Lecture 10: The Law of Iterated Logarithm, Charles University](https://kam.mff.cuni.cz/~klazar/l10L_PRTE_20.pdf)
11. [On the law of the iterated logarithm, CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=988528df1587305e47ced19111113f6be9c7795e&repid=rep1&type=pdf)
12. [MA4F7 Brownian motion Lecture Notes, University of Warwick](https://warwick.ac.uk/fac/sci/masdoc/people/studentpages/students2013/egginton/bm_lec_notes2.pdf)
13. [Brownian motion chapter, financial mathematics notes (S. Lee)](https://github.com/SungchulLee/financial_math_book_writing/blob/main/docs/ch02/brownian_motion/brownian_motion.md)
14. [On the law of the iterated logarithm under the sub-linear expectations, L.-X. Zhang, arXiv:2103.01390](https://arxiv.org/abs/2103.01390)
15. [Confidence sequences for mean, variance, and median, D. A. Darling and H. Robbins, PNAS 58(1):66–68 (1967)](https://pmc.ncbi.nlm.nih.gov/articles/PMC335597/)
16. [A nonasymptotic law of iterated logarithm for general M-estimators, Schreuder, Brunel, Dalalyan, AISTATS 2020](https://proceedings.mlr.press/v108/dalalyan20a/dalalyan20a.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Continuous-time and continuous-state processes › Gaussian and Wiener processes › Brownian sample-path fine structure*

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