Kolmogorov's zero–one law

For an independent sequence of random variables, every event in the tail σ-algebra has probability zero or one. The tail σ-algebra contains events unchanged by altering any finite number of variables.

Tail σ-algebra: The σ-algebra of events measurable from every suffix of a sequence, unchanged by altering finitely many terms. Its events are exactly those to which Kolmogorov's law applies.
Probability space: A measure space whose points are outcomes and whose measure assigns probabilities to measurable events. The zero-or-one conclusion is a statement about probabilities within this structure.
Borel–Cantelli lemmas: Theorems relating the probabilities of events to whether infinitely many of them occur. Independence can make the event of infinitely many occurrences a tail event with probability zero or one.
Hewitt–Savage zero–one law: A zero–one law for events invariant under finite permutations of an independent identically distributed sequence. It uses identical distributions and permutation invariance, rather than invariance under all finite changes.
Independence (probability theory): A property whereby the occurrence of one event does not change the probabilities of others. Independence of the sequence makes each tail event independent of every finite initial segment.
Random variable: A measurable function from a probability space to a measurable value space. The law starts with a sequence of these measurable functions.
Convergence of random series: The study of whether a series of random variables converges, often almost surely or in probability. Convergence is unchanged by finitely many terms, so it can be a tail event.
Blumenthal's zero–one law: A theorem giving probabilities zero or one to events in the initial σ-algebra of a process with trivial germ information. It concerns information near time zero, whereas Kolmogorov's law concerns information in the remote tail.
σ-algebra: A collection of sets closed under complements and countable unions, used to define measurable events. The tail σ-algebra is a particular σ-algebra, and the law concerns probabilities of its events.
Generated σ-algebra: The smallest σ-algebra making a specified collection of functions or sets measurable. Initial segments and suffixes generate the σ-algebras used to define tail events.

Linked from 2 pages

Andrey KolmogorovBroader topic: It captures how distant events become deterministic in probability despite underlying randomness.