Let $\{ X_{n, m} \}_{n \geq 1, \, 1 \leq m \leq M_n}$ be a
D5164: Random real standard triangular array such that
| (i) |
$X_{n, 1}, \dots, X_{n, n}$ is an D2713: Independent random collection for each $n \geq 1$
|
| (ii) |
$\mu_{n, m} : = \mathbb{E} X_{n, m} \in \mathbb{R}$
|
| (iii) |
$\sigma^2_{n, m} : = \text{Var} X_{n, m} \in (0, \infty)$
|
| (iv) |
\begin{equation}
s^2_n : = \sum_{m = 1}^{M_n} \sigma^2_{n, m}
\end{equation}
|
| (v) |
\begin{equation}
Z_{n, m}
: = \frac{X_{n, m} - \mu_{n, m}}{s_n}
\end{equation}
|
Since $X_{n, 1}, \ldots, X_{n, M_n}$ are independent for each $n \geq 1$, applying results
one has
\begin{equation}
\begin{split}
\sum_{m = 1}^{M_n} \text{Var}(Z_{n, m})
& = \sum_{m = 1}^{M_n} \text{Var} \left( \frac{X_{n, m} - \mu_{n, m}}{s_n} \right) \\
& = \sum_{m = 1}^{M_n} \text{Var} \left( \frac{X_{n, m}}{s_n} \right) \\
& = \frac{1}{s^2_n} \sum_{m = 1}^{M_n} \text{Var} ( X_{n, m} ) \\
& = \frac{1}{s^2_n} \sum_{m = 1}^{M_n} \sigma^2_{n, m} \\
& = \frac{1}{s^2_n} s^2_n \\
& = 1
\end{split}
\end{equation}
Finally,
R3791: Linearity of signed basic expectation implies
\begin{equation}
\mathbb{E} (Z_{n, m})
= \mathbb{E} (X_{n, m} - \mu_{n, m})
= \mathbb{E} (X_{n, m}) - \mu_{n, m}
= \mu_{n, m} - \mu_{n, m}
= 0
\end{equation}
$\square$
