Convergence in law

Definition
Let:

• a sequence of random variables $(X_n)$ with $(F_n)$ the corresponding sequence of distribution functions
•  a random variable $X$ of distribution function $F$

 $(X_n)$ converge in law to $X$ (denoted as $X_n \mathop{\stackrel{\mathcal{L}}{\longrightarrow}}  X$) if $$ \forall x \in \mathcal{C}(F): \lim_{n \rightarrow \infty} F_n(x) = F(x) $$

Example:
Let $X_n = \frac{1}{\sqrt{n}} \frac{\sum_i^n{Z_i-n}}{\sqrt{2}} $ with $Z_i$ i.i.d. $\mathcal{X}_1^2$ random variables. $$ X_n  \mathop{\stackrel{\mathcal{L}}{\longrightarrow}} \mathcal{N}(0, 1) $$

R code:

 require(ConvergenceConcepts)  
 rand <- function(n){(cumsum(rchisq(n,df=1))-(1:n))/sqrt(2*(1:n))}  
 data <- generate(randomgen=rand,nmax=500,M=5000)$data  
 law.plot2d(data)

 law.plot3d(data,pnorm)