확률수렴(Convergence in probability)

Def)
$X_n \overset{p}{\rightarrow} X$ <=> $\forall \epsilon >0, \displaystyle \lim_{n \to \infty} P(|X_n-X| \geq \epsilon) = 0$

WLLN(Weak Law of Large Number)

For iid random sample $X_1,...,X_n$
$\overline{X_n} \overset{p}{\rightarrow} \mu$ if $\mu =E(X_1)$ & $Var(X_1) < \infty$
pf)

Continuous Mapping Theorem

If $X_n \overset{p}{\rightarrow} X$ & $Y_n \overset{p}{\rightarrow} Y$, then $g(X_n,Y_n) \overset{p}{\rightarrow} g(X,Y)$ for any continuous function $g$.

ex1)
If $\overline{X_n} \overset{p}{\rightarrow} \mu_X, \overline{Y_n} \overset{p}{\rightarrow} \mu_Y$, then
$3(log \overline{X_n})^{\overline{Y_n}} \overset{p}{\rightarrow} 3(log \space \mu_X)^{\mu_Y}$