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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. Let F {\displaystyle {\mathcal {F}}} be the σ-algebra generated by the X i {\displaystyle X_{i}} . Then, a tail event F ∈ F {\displaystyle F\in {\mathcal {F}}} is an event which is probabilistically independent of each finite subset of these random variables. 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.
Tail events are precisely those events whose occurrence can still be determined if an arbitrarily large but finite initial segment of the X i {\displaystyle X_{i}} is removed.