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Kolmogorov's zero-one law

From Academic Kids

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

<math>X_1,X_2,X_3,\dots\,<math>

is an infinite sequence of independent random variables (not necessarily identically distributed). Then, 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

<math>\sum_{k=1}^\infty X_k<math>

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.

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: since there is a positive, though tiny, chance that the monkey "gets it right" the first time he tries, the probability of the tail event that he "gets it right" given an infinite amount of time cannot be zero. Therefore, that probability must be 1 by the zero-one law. (The independence of the individual events -- keystrokes -- is understated in this particular example, i.e. the monkey doesn't get bored, or makes a fixation on a particular key, etc.)

External links

  • The Legacy of Andrei Nikolaevich Kolmogorov (http://www.kolmogorov.com/) Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
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