Law of the iterated logarithm

A family of probability theorems describing the almost-sure upper and lower fluctuations of sums of independent random variables, typically on the scale √(n log log n).

Hartman–Wintner law of the iterated logarithm: A classical iterated-logarithm theorem for independent, identically distributed variables with mean zero and finite positive variance. It gives the standard √(2n log log n) fluctuation scale under finite-variance assumptions.
Independent random variables: Random variables whose joint probabilities factor into products of their individual probabilities. Independence is the main structural assumption in the classical forms of the theorem.
Alexander Khintchine: A Soviet mathematician whose work included foundational results in probability theory and analysis. Khintchine established an early iterated-logarithm result for sums of independent random variables.
Simple random walk: A process that moves by independent steps, commonly one unit left or right with equal probability. Its position after n steps fluctuates on the iterated-logarithm scale almost surely.
Strong law of large numbers: A theorem stating that sample averages converge almost surely to their expected value under specified conditions. It gives convergence of averages but not the precise size of their largest residual fluctuations.
Kolmogorov law of the iterated logarithm: A version of the iterated logarithm law for independent, non-identically distributed variables under suitable variance and tail conditions. It extends the classical result beyond identically distributed summands.
Partial sum: The sum of the first several terms of a sequence. The theorem tracks the running sums whose fluctuations grow with the number of terms.
Andrey Kolmogorov: A Russian mathematician who established modern foundations for probability theory. Kolmogorov proved general forms of the law under conditions allowing non-identical summands.
Empirical distribution function: The fraction of observations in a sample that fall at or below each possible value. Iterated-logarithm results describe almost-sure fluctuations of empirical distributions around their population values.
Functional central limit theorem: A convergence theorem for rescaled stochastic processes to Brownian motion or another limiting process. It gives distributional process limits, whereas the iterated logarithm law asserts pathwise extreme bounds.

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John Edensor LittlewoodRelated: Littlewood proved a general form that established the law’s sharp fluctuation scale.