Good morning,
I'm having some trouble understanding the weak large number theorem that says:
Let $(X_n)_{n\in{\mathbb{N}}}:\Omega \to \mathbb{R}$ be a sequence of real random variables with $X_n\in{L^2}$. Let $X_n$ have the same mean $E[X_n]=m$ $\forall{n}$ and have bounded variances $\exists c\in{\mathbb{R}}:\forall{n}\,Var(X_n)\leq{c}$. Let $Cov(X_n,X_m)=0\,\forall{n\neq{m}}$. Let $S_n=\sum_1^n X_n$. Then $\frac{S_n}{n}\to m$ in probability.
My problem is to understand what $S_n$ is exactly. $X_n$ are functions and I don't know how I can build them in practice. For example, let's consider the roll of a die. In this case $\Omega$ should be $\Omega=\{w=(w_1,w_2,w_3,...)|w_i\in{\{1,2,3,4,5,6\}}\}$. In practice $w_i$ is the upper face of the die at the $i$th throw. If at the first throw the upper face is $1$ then I can build $X_1$ in this way: $$ X_1(w)= \begin{cases} 1 & if\,w_1=1 \\ 0 & otherwise \\ \end{cases}$$ If at the second throw the upper face is $4$ then Should I build $X_2$ in this way: $$ X_2(w)= \begin{cases} 1 & if\,w_2=4 \\ 0 & otherwise \\ \end{cases}$$ or in this way? $$ X_2(w)= \begin{cases} 1 & if\,w_1=1\land w_2=4 \\ 0 & otherwise \\ \end{cases}$$ So $S_n$ is a new random variable that is defined in this way. If I choose the first definition of $X_2$ I have: $$ S_n(w)=X_1(w)+X_2(w)= \begin{cases} 1 & if\,(w_1=1\land w_2\neq4)\lor(w_1\neq1\land w_2=4) \\ 2 & if\,w_1=1\land w_2=4\\ 0 & otherwise \\ \end{cases}$$ If I choose the second definition of $X_2$ I have: $$ S_n(w)=X_1(w)+X_2(w)= \begin{cases} 1 & if\,w_1=1\land w_2\neq4 \\ 2 & if\,w_1=1\land w_2=4\\ 0 & otherwise \\ \end{cases}$$ With this building, it seems to me that $S_n$ makes sense. But how can I calculate the mean value of such functions $X_n$?
In reality, after the theorem, there is an example in my notes. It says this:
Let's consider rolling a die. We have a sequence of random variables $X_1$, $X_2$, ..., that have the same uniform probability law on $\{1,2,3,4,5,6\}$. We can calculate that $E[X_n]=\frac{7}{2}$.
I think that it is calculated in this way: $E[X_n]=\frac{1+2+3+4+5+6}{6}$. It seems that $X_n: \{1,2,3,4,5,6\}\to \{1,2,3,4,5,6\}$ with $X(w)=w$. In fact for a discrete random variable $E[X]$ can be calculated in this way: $E[X]=\sum_{w\in{\Omega}}X(w)\mathbb{P}(w)$ and for uniform probability on $\Omega=\{1,2,3,4,5,6\}$ I have that $\mathbb{P}(w)=\frac{1}{6}$.
But then $S_n=\sum_i^n X_i=nX_1$ and this is nonsense to me because I don't think the theorem is talking about this $S_n$.
I have also another question. I don't understand how a random variable can be expressed formulaically in the continuous case. For example, if I measure the length of a table, the date should distribute as a gaussian. But what is $\Omega$ in this case and what is the function $X$?
Thanks for your availability.