Hewitt–Savage zero-one law

The Hewitt–Savage zero–one law is a theorem in probability theory, similar to Kolmogorov's zero–one law and the Borel–Cantelli lemma, that specifies that a certain type of event will either almost surely happen or almost surely not happen. It is sometimes known as the Hewitt–Savage law for symmetric events. It is named after Edwin Hewitt and Leonard Jimmie Savage.

Let ${\displaystyle \left\{X_{n}\right\}_{n=1}^{\infty }}$ be a sequence of independent and identically-distributed random variables taking values in a set ${\displaystyle \mathbb {X} }$. The Hewitt–Savage zero–one law says that any event whose occurrence or non-occurrence is determined by the values of these random variables and whose occurrence or non-occurrence is unchanged by finite permutations of the indices, has probability either 0 or 1 (a "finite" permutation is one that leaves all but finitely many of the indices fixed).

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