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 Savage-Hewitt law for symmetric events. It is named after Edwin Hewitt and Leonard Jimmie Savage.
Statement of the Hewitt-Savage zero-one law
Let be a sequence of independent and identically distributed random variables taking values in a set. 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.Somewhat more abstractly, define the exchangeable sigma algebra or sigma algebra of symmetric events to be the set of events which are invariant under finite permutations of the indices in the sequence. Then.
Since any finite permutation can be written as a product of transpositions, if we wish to check whether or not an event is symmetric, it is enough to check if its occurrence is unchanged by an arbitrary transposition,.
Example
Let the sequence of independent and identically distributed random variables taking values in. Consider the random walk. Then one of the following occurs with probability 1:- and.
First consider the case when X1 is a.s. constant. Then with probability 1 we have that either, or.
Now consider the case, when X1 is not a.s. constant. Then for any the event is in the exchangeable sigma algebra. That is because limit supremum does not change with finite permutation of the indices. From Hewitt-Savage zero-one law we have that
There has to exist t, where probability switches from 0 to 1 i.e. exists such that almost surely. Similarly exists such that almost surely.
Since almost surely
and X1 is not a.s. 0, then is not finite. Similarly in not finite.
Therefore, with probability 1 either,
or.