Kolmogorov's three-series theorem

A criterion for almost-sure convergence of a series of independent random variables, expressed through tail probabilities and the means and variances of suitably truncated variables.

Truncation (probability theory): A procedure that replaces values outside a chosen range with a boundary value or discards them. The theorem tests bounded versions of each random variable at a fixed cutoff.
Independent random variables: Random variables whose joint probabilities factor into products of their marginal probabilities. Independence is the theorem's central hypothesis and enables its fluctuation bounds.
Andrey Kolmogorov: A Russian mathematician who established foundational results in probability theory and measure theory. The theorem bears his name and reflects his systematic approach to probability.
Random series: An infinite sum whose terms are random variables. The theorem supplies a direct convergence test for this class of sums.
Borel–Cantelli lemma: A result relating sums of event probabilities to whether infinitely many of those events occur. Its first lemma controls how often terms exceed the truncation threshold.
Truncated expectation: The expectation of a random variable after restricting or clipping its values at a threshold. The theorem requires the series of means of threshold-truncated terms to converge.
Probability theory: The mathematical study of random events, variables, and their distributions. The theorem is a foundational convergence result within this broader discipline.
Rademacher series: A random series whose terms are fixed coefficients multiplied by independent symmetric signs. Its convergence can often be reduced to square-summability of the coefficients.
Kolmogorov's maximal inequality: An inequality bounding the probability that partial sums of independent mean-zero variables exceed a fixed magnitude. It bounds fluctuations of centered truncated sums using their variances.
Variance: A measure of the average squared deviation of a random variable from its mean. The theorem sums variances of the truncated variables to control accumulated fluctuation.

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