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.