Zero-One Laws

There are several common zero-one laws which identify the possible range of a random variable to be trivial. There are also several zero-one laws which provide the basis for all proofs of almost sure convergence.

Proposition 1. Borel-Cantelli Lemma Let \(\{A_n\}\) be any events (not necessarily independent).

If \(\sum_n{P(A_n)}<\infty\), then
\[P([A_n \; i.o.])=P(\underset{n\rightarrow \infty}{\text{lim sup}} A_n) = 0\].
Proposition 2. Borel Zero-One Law If \(\{A_n\}\) is a sequence of independent events, then

\[  \begin{equation*}
P([A_n \; i.o.])= \begin{cases}
0, \quad &  \text{iff} \sum_n P(A_n) < \infty \\
1, \quad &  \text{iff} \sum_n P(A_n) = \infty
\end{cases}
\end{equation*}
\]
Definition.  An almost trivial \(\sigma\)-field is a \(\sigma\)-field all of whose events has probability 0 or 1.

Theorem 3. Kolmogorov Zero-One Law If \(\{X_n\}\) are independent random variables with tail \(\sigma\)-field \(\mathcal{T}\), then \(\Lambda\in \mathcal{T}\) implies \(P(\Lambda)=0\) or 1 so that the tail \(\sigma\)-field is almost trivial.

Lemma 4. Almost trivial \(\sigma\)-fields  Let \(\mathcal{G}\) be an almost trivial \(\sigma\)-field and let \(X\) be a random variable measurable with respect to \(\mathcal{G}\).  Then there exists \(c\) such that \(P[X=c] = 1\).

Corollary 5.  Let \(\{X_n\}\) be independent random variables.  Then the following are true.

(a) The event
\[ [\sum_n X_n \;converges] \] has probability 0 or 1.

(b) The random variables \(\text{lim sup}_{n\rightarrow \infty}X_n\) and \(\text{lim inf}_{n\rightarrow \infty}X_n\) are constant with probability 1.

(c) The event
\[ \{\omega: S_n(\omega)/n \rightarrow 0 \} \] has probability 0 or 1.