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Encyclopedia > Kolmogorov's zero-one law

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Kolmogorov's zero-one law

In probability theory, Kolmogorov's zero-one law, named in honor of Andrey Nikolaevich Kolmogorov, treats of probabilities of certain "tail events" defined in terms of infinite sequences of random variables. Suppose
<math>X_1,X_2,X_3,\dots</math>
is an infinite sequence of independent random variables (not necessarily identically distributed). A tail event is an event whose occurrence or failure depends on the values of the random variables in this sequence and on nothing else, but which is probabilistically independent of each finite subsequence of this sequence of random variables. For example, the event that the series
<math>\sum_{k=1}^\infty X_n</math>
converges, is a tail event. In an infinite sequence of coin-tosses, the probability that a sequence of 100 consecutive heads eventually occurs, is a tail event.

Kolmogorov's zero-one law states that the probability of any tail event is either zero or one.

In a book published in 1909, Emile Borel[?] stated that if a dactylographic monkey hits typewriter keys randomly forever, it will eventually type every book in France's National Library. That is a special case of this zero-one law.



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