Kolmogorov's zero-one law |
In probability theory, Kolmogorov's zero-one
law, named in honor of Andrey
Nikolaevich Kolmogorov, deals with probabilities of certain "tail events" defined in terms of infinite sequences of random variables. Suppose
-
is an infinite sequence of independent random
variables (not necessarily identically distributed). A tail event is an event whose occurrence or failure is
determined by the values of these random variables but which is probabilistically independent of each finite subsequence of these random variables. For example,
the event that the series
-
converges, is a tail event. The event that the sum to which it converges is more than 1 is not a tail event, since,
for example, it is not independent of the value of X1. 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, Émile
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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