Hewitt–Savage zero–one law

Template:Short description 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 ${\left\{X_n\right\}}_{n=1}^{\mathrm{\infty }}$ be a sequence of independent and identically distributed random variables taking values in a set $𝕏$. 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).

Somewhat more abstractly, define the exchangeable sigma algebra or sigma algebra of symmetric events $ℰ$ to be the set of events (depending on the sequence of variables ${\left\{X_n\right\}}_{n=1}^{\mathrm{\infty }}$) which are invariant under finite permutations of the indices in the sequence ${\left\{X_n\right\}}_{n=1}^{\mathrm{\infty }}$. Then $A\in ℰ⟹\mathbb{P}(A)\in \{0,1\}$.

Since any finite permutation can be written as a product of transpositions, if we wish to check whether or not an event $A$ is symmetric (lies in $ℰ$), it is enough to check if its occurrence is unchanged by an arbitrary transposition $(i,j)$, $i,j\in \mathbb{N}$.

Example

Let the sequence ${\left\{X_n\right\}}_{n=1}^{\mathrm{\infty }}$ of independent and identically distributed random variables taking values in $ℝ$. Consider the random walk $S_N=\sum _{n=1}^N{X}_n$. Then one of the following occurs with probability 1:

• $S_N=0$
• $S_N\to \mathrm{\infty }$
• $S_N\to -\mathrm{\infty }$
• $lim\; inf{S}_N=-\mathrm{\infty }$ and $lim\; sup{S}_N=\mathrm{\infty }$.

Since S_N are not independent the Kolmogorov's zero–one law is not directly applicable.

First consider the case when X₁ is a.s. constant. Then with probability 1 we have that either ($S_N=0$), ($S_N\to \mathrm{\infty }$) or ($S_N\to -\mathrm{\infty }$).

Now consider the case, when X₁ is not a.s. constant. Then for any $t\in [-\mathrm{\infty },\mathrm{\infty }]$ the event $\{lim\; sup{S}_N\ge t\}$ 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

$\mathbb{P}(lim\; sup{S}_N\ge t)\in \{0,1\}$.

There has to exist t, where probability switches from 0 to 1 i.e. exists $t^{\ast }\in [-\mathrm{\infty },\mathrm{\infty }]$ such that $lim\; sup{S}_n=t^{\ast }$ almost surely. Similarly exists $t_{\ast }\in [-\mathrm{\infty },\mathrm{\infty }]$ such that $lim\; inf{S}_n=t_{\ast }$ almost surely.

Since almost surely

$t^{\ast }=lim\; sup{S}_N=X_1+lim\; sup\sum _{n=2}^N{X}_n=X_1+t^{\ast }$

and X₁ is not a.s. 0, then $t^{\ast }$ is not finite. Similarly $t_{\ast }$ in not finite.

Therefore, with probability 1 either ($S_N\to \mathrm{\infty }$), ($S_N\to -\mathrm{\infty }$) or ($lim\; inf{S}_N=-\mathrm{\infty }$ and $lim\; sup{S}_N=\mathrm{\infty }$).