Convergence in distribution

Def)
$X_n \overset{d}{\rightarrow} X <=> \forall x, \displaystyle \lim_{ n \to \infty} F_{X_n}(x)=F_X(x)$

2 useful theorem)
① Continuous Mapping theorem
$X_n \overset{d}{\rightarrow} X => g(X_n) \overset{d}{\rightarrow} g(X)$ for any continuous function $g$

② Slutsky's theorem
If $X_n \overset{d}{\rightarrow} X, Y_n \overset{p}{\rightarrow} c_1,Z_n \overset{p}{\rightarrow} c_2(c_1,c_2: constant),$ then $Y_nX_n+Z_n \overset{d}{\rightarrow} c_1X+c_2$

ex1) $X_1,...,X_n \overset{iid}{\sim} N(\mu, \sigma^2)$
Let $T_n=\frac{\overline{X_n} - \mu}{s/ \sqrt{n}}$ where $s=\sqrt{\frac{\sum_{i} (X_i-\overline{X_n})^2}{n-1}}$, then $T_n \overset{d}{\rightarrow} Z$ where $Z \sim N(0,1)$

pf)
$\overline{X_n} \sim N(\mu, \frac{\sigma^2}{n}) => \frac{\overline{X_n} - \mu}{ \sigma / \sqrt{n}} \sim N(0, 1) \\ T_n = \frac{\overline{X_n} - \mu}{\sigma / \sqrt{n}} \times \frac{\sigma}{s} \overset{d}{\rightarrow} Z$ by slutsky's theorem $(\frac{\sigma}{s} \overset{p}{\rightarrow} 1)$

CLT(Central Limit Theorem)

Suppose $X_1,...,X_n$ is a random sample from a distribution having mean:0 & variance:1, then $Y_n=\sqrt{n} \space \overline{X_n} \overset{d}{\rightarrow} N(0,1)$

pf)

Generalization)
When $X_1,...,X_n$ is a random sample from a distribution having mean: $\mu$ & variance: $\sigma^2$,
$\frac{\sqrt{n}(\overline{X_n} - \mu)}{\sigma} \overset{d}{\rightarrow} N(0,1)$

pf)

Delta method

If $\sqrt{n}(X_n - \theta) \overset{d}{\rightarrow} N(0,\sigma^2)$, then
$\sqrt{n}(g(X_n) - g(\theta)) \overset{d}{\rightarrow} N(0,\sigma^2g'^2(\theta))$
for $g() \space s.t. g''(\theta)<\infty,g'(\theta) \neq 0$

pf)

ex1) $X_1,...,X_n \overset{iid}{\sim} Gamma(1,\lambda)$
$\sqrt{n}(log \overline{X_n}-?) \overset{d}{\rightarrow} N(0,?) \\ E(X_1)=\lambda, Var(X_1)=\lambda^2, E(\overline{X_n})=\lambda, Var(\overline{X_n})=\frac{\lambda^2}{n} \\ \sqrt{n}(\overline{X_n} - \lambda) \overset{d}{\rightarrow} N(0,\lambda^2)$ by CLT
$g(\overline{X_n})=log \overline{X_n} => g'(\overline{X_n}) = 1/\overline{X_n} \\ \sqrt{n}(log \overline{X_n} - log \lambda) \overset{d}{\rightarrow} N(0,\frac{\lambda^2}{n} \times \frac{1}{\lambda^2})$