Borel–Cantelli lemma

Definition:
Let a sequence  of random variables $(X_n)$:

1. If $\sum_{n  = 1}^{\infty}  Pr(X_n) < \infty$ then $$ Pr(\limsup_{n \rightarrow \infty}  X_n) = Pr(X_n  infinitly  often) = 0$$
2. If  $\sum_{n  = 1}^{\infty}  Pr(X_n) = \infty$ and the events $X_n$ are independent then $$ Pr(\limsup_{n \rightarrow \infty}  X_n) = Pr(X_n  infinitly  often) = 1$$

Limsup/Liminf of a sequence of random variables

Definition:
Let a sequence  of random variables $(X_n)$:
$$\limsup_n  X_n = \cap_p \cup_{n \geq p} X_n $$
\begin{align}

x \in \limsup_n  X_n & \Leftrightarrow \forall  p,  \exists  n \geq p  such  as  x \in X_n & 1\\
& \Leftrightarrow x \in  to  an  infinity  of  X_n &
\end{align}

1. if $ x \in \limsup_n  X_n $ then $\forall  p,  x \in \cup_{n \geq p}  X_n$, for every $p$ you will find an $X_n$ with $n > p$ of which $x$ is a member.

Similarly:

$$\liminf_n  X_n = \cup_p \cap_{n \geq p} X_n $$
\begin{align}

x \in \liminf_n  X_n & \Leftrightarrow  \exists  p,  \forall  n \geq p,  x \in X_n & \\
& \Leftrightarrow  x \in X_n  except  a  finite  number  of  indices  n &
\end{align}

Example:

Let:

• $X_1 = \emptyset$
• $X_2 = [-3, 3]$
• $X_{2n - 1} = [-1, 2[$, $X_{2n} = [-2, 1]$ with $n \geq 1$

\begin{align}
\limsup_{n \rightarrow \infty}  X_n & = [-2, 2[ \\
\liminf_{n \rightarrow \infty}  X_n  & = [-1, 1]
\end{align}

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)