Kolmogorov's zero–one law

In probability theory, Kolmogorov's zero–one law, named in honor of Andrey Nikolaevich Kolmogorov, specifies that a certain type of event, called a tail event, will either almost surely happen or almost surely not happen; that is, the probability of such an event occurring is zero or one.

Tail events are defined in terms of infinite sequences of random variables. Suppose

is an infinite sequence of independent random variables (not necessarily identically distributed). Let ${\displaystyle {\mathcal {F}}}$ be the σ-algebra generated by the ${\displaystyle X_{i}}$. Then, a tail event ${\displaystyle F\in {\mathcal {F}}}$ is an event which is probabilistically independent of each finite subset of these random variables. (Note: ${\displaystyle F}$ belonging to ${\displaystyle {\mathcal {F}}}$ implies that membership in ${\displaystyle F}$ is uniquely determined by the values of the ${\displaystyle X_{i}}$ but the latter condition is strictly weaker and does not suffice to prove the zero-one law.) For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event.

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