Limsup/Liminf of a sequence of random variables

Definition:
Let a sequence  of random variables $(X_n)$:
$$\limsup_n  X_n = \cap_p \cup_{n \geq p} X_n $$
\begin{align}

x \in \limsup_n  X_n & \Leftrightarrow \forall  p,  \exists  n \geq p  such  as  x \in X_n & 1\\
& \Leftrightarrow x \in  to  an  infinity  of  X_n &
\end{align}

1. if $ x \in \limsup_n  X_n $ then $\forall  p,  x \in \cup_{n \geq p}  X_n$, for every $p$ you will find an $X_n$ with $n > p$ of which $x$ is a member.

Similarly:

$$\liminf_n  X_n = \cup_p \cap_{n \geq p} X_n $$
\begin{align}

x \in \liminf_n  X_n & \Leftrightarrow  \exists  p,  \forall  n \geq p,  x \in X_n & \\
& \Leftrightarrow  x \in X_n  except  a  finite  number  of  indices  n &
\end{align}

Example:

Let:

• $X_1 = \emptyset$
• $X_2 = [-3, 3]$
• $X_{2n - 1} = [-1, 2[$, $X_{2n} = [-2, 1]$ with $n \geq 1$

\begin{align}
\limsup_{n \rightarrow \infty}  X_n & = [-2, 2[ \\
\liminf_{n \rightarrow \infty}  X_n  & = [-1, 1]
\end{align}