矩生成函数 中心极限定理

hbghlyj

mgf_color.pdf
Problem 6.5.8. Let $Y_1, Y_2, \ldots, Y_n$ be independent random variables each with the Bernoulli $B(p)$ distribution, for some $p \in(0,1)$.

• Show that the mgf $m_{W_n}(t)$ of the random variable
  \[
  W_n=\frac{Y_1+Y_2+\cdots+Y_n-n p}{\sqrt{n p(1-p)}},
  \]
  can be written in the form
  \[
  m_{W_n}(t)=\left(p e^{\frac{t}{\sqrt{n}} \alpha}+(1-p) e^{-\frac{t}{\sqrt{n}} \alpha^{-1}}\right)^n,\tag{6.5.1}
  \]
  for some $\alpha$ and find its value.
• Write down the Taylor approximations in $t$ around 0 for the functions $\exp \left(\frac{t}{\sqrt{n}} \alpha\right)$ and $\exp \left(-\frac{t}{\sqrt{n}} \alpha^{-1}\right)$, up to and including the term involving $t^2$. Then, substitute those approximations in (6.5.1) above. What do you get? When $n$ is large, $\frac{t}{\sqrt{n}} \alpha$ and $\frac{t}{\sqrt{n}} \alpha^{-1}$ are close to 0 and it can be shown that the expression you got is the limit of $m_{W_n}(t)$, as $n \rightarrow \infty$.
• What distribution is that limit the mgf of?

hbghlyj

尝试写一下...

• \begin{array}{l}m_{W_n}(t)\\=\mathbb{E}\left(\exp \left(\frac{t\left(Y_1+Y_2+\cdots+Y_n-np\right)}{\sqrt{np(1-p)}}\right)\right) \\ =\mathbb{E}\left(\exp \left(\frac{t\left(Y_{1}-p\right)}{\sqrt{np(1-p)}}\right) \ldots \exp \left(\frac{t\left(Y_{n}-p\right)}{\sqrt{np(1-p)}}\right)\right)\\
  =\left[\mathbb{E}\left(\exp \left(\frac{t\left(Y_{1}-p\right)}{\sqrt{np(1-p)}}\right)\right)\right]^n\\
  =\left[p\exp \left(\frac{t\left(1-p\right)}{\sqrt{np(1-p)}}\right)+(1-p)\exp \left(\frac{-tp}{\sqrt{np(1-p)}}\right)\right]^n
  \end{array}所以$\alpha=\sqrt{1-p\over p}$.
• 将\begin{array}l
  \exp\left(\frac t{\sqrt n}\alpha\right)=1+\frac{\alpha  t}{\sqrt{n}}+\frac{\alpha ^2 t^2}{2 n}+O\left(t^3\right)\\
  \exp\left(-\frac t{\sqrt n}\alpha^{-1}\right)=1-\frac{t}{\alpha  \sqrt{n}}+\frac{t^2}{2 \alpha ^2 n}+O\left(t^3\right)\end{array}代入(6.5.1),由$\alpha ^2=\frac{1-p}p$得
  $t^0$系数$p+(1-p)=1$
  $t^1$系数$p\frac{\alpha t}{\sqrt n}+(1-p)\left(-\frac t{\alpha\sqrt n}\right)=0$
  $t^2$系数$p\frac{\alpha^2 t^2}{2n}+(1-p)\frac{t^2}{2\alpha^2n}=\frac{t^2}{2n}$
  得到\[m_{W_n}(t)=\left[1+\frac{ t^2}{2 n}+O\left(t^3\right)\right]^n\to\exp\left(t^2\over2\right)\]
• 根据这个pdf第二页, the last line is the mgf of $N(0, 1)$.
  Using the continuity theorem, $W_n$依分布收敛到$N(0,1)$.