Let $X_1, X_2, X_3, \ldots \in \text{Random}(\Omega \to \mathbb{R})$ each be a
D3161: Random real number such that
| (i) |
$X_1, X_2, X_3, \ldots$ is an D3358: I.I.D. random collection
|
| (ii) |
\begin{equation}
\exists \, \mu \in \mathbb{R} :
\mathbb{E} X_1 = \mu
\end{equation}
|
| (iii) |
\begin{equation}
\mathbb{E} |X_1|^2 < \infty
\end{equation}
|
Then
| (1) |
\begin{equation}
\frac{1}{N} \sum_{n = 1}^N X_n \overset{L^2}{\longrightarrow} \mu \quad \text{ as } \quad N \to \infty
\end{equation}
|
| (2) |
\begin{equation}
\frac{1}{N} \sum_{n = 1}^N X_n \overset{p}{\longrightarrow} \mu \quad \text{ as } \quad N \to \infty
\end{equation}
|
