5-27 The Central Limit Theorem
In this section,
- $N$,
- a r.v. with standard normal distribution
Theorem 27.1
- $c \in \mathbb{R}^{n}$,
- $\sigma > 0$,
- \(\{X_{n}\}_{n \in \mathbb{N}}\),
- independent sequence of r.v.s with mean $c$ and $\sigma^{2}$
If
\[S_{n} := \sum_{i=1}^{n} X_{i} ,\] \[\frac{ S_{n} - nc }{ \sigma \sqrt{n} } \overset{d}{\rightarrow} N ,\]$\mu$ is said to be complete if
\[N \in \mathcal{A}, \ \mi(N) = 0, \ \Rightarrow \ \forall S \subseteq N, \ S \in \mathcal{A}\]■
Lemma 1
- $z_{1}, \ldots, z_{m}$,
- complex numbers of moduls at most 1
- $w_{1}, \ldots, z_{m}$,
- complex numbers of moduls at most 1
proof
Let $m = 2$.
\[\begin{eqnarray} (z_{1} - w_{1}) z_{2} + w_{1} (z_{2} - w_{2}) & = & z_{1}z_{2} - w_{1}z_{2} + w_{1}z_{2} - z_{1}z_{2} \nonumber \\ & = & z_{1}z_{2} - w_{1}w_{2} \nonumber \end{eqnarray}\]Thus,
\[\begin{eqnarray} \abs{ z_{1}z_{2} - w_{1}w_{2} } & \le & \abs{ (z_{1} - w_{1}) z_{2} } + \abs{ w_{1} (z_{2} - w_{2}) } \nonumber \\ & \le & \abs{ (z_{1} - w_{1}) } + \abs{ (z_{2} - w_{2}) } . \nonumber \end{eqnarray}\]Suppose that the statement holds up to $m = n - 1$. Then we prove the statement in case of $m = n$.
\[\begin{eqnarray} (z_{1} - w_{1}) z_{2}\cdots z_{m} + w_{1} (z_{2} \cdots z_{m} - w_{2} \cdots w_{m}) & = & z_{1}\cdots z_{m} - w_{1}z_{2} \cdots z_{m} + w_{1}z_{2} \cdots z_{M} - w_{1}w_{2} \cdots w_{m} \nonumber \\ & = & z_{1} \cdots z_{m} - w_{1} \cdots w_{m} \nonumber \end{eqnarray}\]Thus,
\[\begin{eqnarray} \abs{ z_{1} \cdots z_{m} - w_{1} \cdots w_{m} } & \le & \abs{ (z_{1} - w_{1}) z_{2}\cdots z_{m} } + \abs{ w_{1} (z_{2} \cdots z_{m} - w_{2} \cdots w_{m}) } \nonumber \\ & \le & \abs{ (z_{1} - w_{1}) } + \abs{ (z_{2} \cdots z_{m} - w_{2} \cdots w_{m}) } \nonumber \\ & \le & \abs{ (z_{1} - w_{1}) } + \sum_{i=2}^{m} \abs{ (z_{i} - w_{i}) } . \nonumber \end{eqnarray}\]$\Box$