| Version 2 |
Version 1 |
| \title{Tail event}% |
\title{Tail event}% |
| \author{Fernando Sanz Gamiz}% |
\author{Fernando Sanz Gamiz}% |
|
|
| \begin{defn} |
\begin{defn} |
|
Let $\Omega$ be a set and $\mathcal F$ a sigma algebra of subsets
|
Let $\Omega$ be a set and $\mathcal F$ a $\sigma$-algebra of subsets
|
| of $\Omega$. Given the random variables $\{X_n, n \in \N\}$, defined |
of $\Omega$. Given the random variables $\{X_n, n \in \N\}$, defined |
| on the measurable space $(\Omega,\mathcal F)$, the \emph{tail |
on the measurable space $(\Omega,\mathcal F)$, the \emph{tail |
| events} are the events of the $\sigma$-algebra |
events} are the events of the $\sigma$-algebra |
| $$\mathcal F_{\infty}=\bigcap^{\infty}_{n=1}\sigma |
$$\mathcal F_{\infty}=\bigcap^{\infty}_{n=1}\sigma |
| (X_n,X_{n+1},\cdots)$$ where $\sigma (X_n,X_{n+1},\cdots)$ is the |
(X_n,X_{n+1},\cdots)$$ where $\sigma (X_n,X_{n+1},\cdots)$ is the |
| $\sigma$-algebra induced by $(X_n,X_{n+1},\cdots)$. |
$\sigma$-algebra induced by $(X_n,X_{n+1},\cdots)$. |
| \end{defn} |
\end{defn} |
|
|
| \medskip |
\medskip |
|
|
| \begin{rem} |
\begin{rem} |
| One can intuitively think of tail events as those events whose |
One can intuitively think of tail events as those events whose |
| ocurrence or not is not affected by altering any finite number of |
ocurrence or not is not affected by altering any finite number of |
| random variables in the sequence. Some examples are $$\{\lim \sup |
random variables in the sequence. Some examples are $$\{\lim \sup |
| X_n <c \}, \sum X_n \mbox{ converges }, \lim X_n \mbox{ exists}$$ |
X_n <c \}, \sum X_n \mbox{ converges }, \lim X_n \mbox{ exists}$$ |
| \end{rem} |
\end{rem} |
|
|
| \medskip |
\medskip |
|
|
| \begin{rem} |
\begin{rem} |
| One of the most important theorems in probability theory due to |
One of the most important theorems in probability theory due to |
| Kolomogorv, is the Kolmogorov zero-one law which states that the |
Kolomogorv, is the Kolmogorov zero-one law which states that the |
| probability of any tail event is 0 or 1 (provided there is a |
probability of any tail event is 0 or 1 (provided there is a |
| probability measure defined on $(\Omega,\mathcal F)$) |
probability measure defined on $(\Omega,\mathcal F)$) |
| \end{rem} |
\end{rem} |
