Glivenko–Cantelli theorem

The theorem states that the empirical distribution function converges uniformly, almost surely, to the population distribution function as the sample size grows.

Empirical distribution function: The fraction of observations at or below each real number, viewed as a function of that number. It is the random function whose uniform distance from the population distribution vanishes.
Cumulative distribution function: A function giving the probability that a random variable is at most each real threshold. The theorem’s population target is this function.
Nonparametric statistics: Statistical methods that do not restrict data to a fixed finite-dimensional family of distributions or models. Uniform empirical convergence supports distribution estimation without a parametric form.
Pointwise convergence: A sequence of functions converges pointwise when it converges at each fixed domain point. The theorem guarantees the stronger uniform convergence, not merely convergence at each threshold.
Uniform convergence: A sequence of functions converges uniformly when its maximum error over the domain tends to zero. The theorem controls the largest discrepancy across all thresholds at once.
Independent and identically distributed random variables: Random variables are independent and identically distributed when they share a distribution and are mutually independent. The classical theorem assumes observations drawn under this common sampling model.
Kolmogorov–Smirnov test: A hypothesis test based on the maximum difference between empirical cumulative distribution functions. Its statistic uses the same supremum discrepancy controlled by the theorem.
Central limit theorem: A theorem describing convergence of suitably normalized sums toward a normal distribution. It gives a distributional approximation for fluctuations rather than almost-sure uniform convergence.
Almost sure convergence: A sequence of random variables converges almost surely when it converges outside an event of probability zero. This specifies the probability-one mode of convergence in the theorem.
Law of large numbers: A family of theorems stating that sample averages converge to their expected values under specified conditions. The result extends pointwise empirical-frequency convergence to all thresholds simultaneously.

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