tail event

Tail eventFernando Sanz Gamiz

Definition.

Let $\mathrm{\Omega }$ be a set and $ℱ$ a sigma algebra of subsetsof $\mathrm{\Omega }$. Given the random variables $\{X_n,n\in \mathbb{N}\}$, definedon the measurable space $(\mathrm{\Omega },ℱ)$, the tailevents are the events of the tail $\sigma$-algebra

$${ℱ}_{\mathrm{\infty }}=\bigcap _{n=1}^{\mathrm{\infty }}\sigma (X_n,X_{n+1},\mathrm{\cdots })$$

where $\sigma (X_n,X_{n+1},\mathrm{\cdots })$ is the$\sigma$-algebra induced by $(X_n,X_{n+1},\mathrm{\cdots })$.

Remark 1.

One can intuitively think of tail events as those events whoseocurrence or not is not affected by altering any finite number ofrandom variables in the sequence. Some examples are

Remark 2.

One of the most important theorems in probability theory due toKolomogorv, is the Kolmogorov zero-one law which states that, in the case of independent random variables, theprobability of any tail event is 0 or 1 (provided there is aprobability measure defined on $(\mathrm{\Omega },ℱ)$)