We consider a sequence $A_n$ of subsets of $\mathbb{Z}$. At the step $n$, a particle is thrown at the origin and is moving until it goes out of $A_n$ by a point $X$. The particle is moving at the right with probability $1/2$, or at the left with probability $1/2$.
We begin with $A_1 = \{0\}$. We note $A_{n+1} = A_n \cup X$.
Example : $A_1 = \{0\}$. At the step $1$, the particle goes out of $A_1$ by the point $-1$ (left) or $1$ (right) ; if it comes out by the right, we note $A_2 = \{0,1\}$. At the step $2$, the particle goes out of $A_1$ by $-1$ (left) or $2$ (right), we adds the point to create $A_3$, etc.
We note $A_n = \{L_n, ..., R_n\}$. $L_n$ is the leftmost point, and $R_n$ is the rightmost point of $A_n$.
We note $Z_n = L_n + R_n$.
I have to simulate $A_n$, $L_n$, $R_n$, $Z_n$ and $p_n = \dfrac{n+2-Z_n}{2(n+2)}$ (when $n$ is big). It's done. I find that $L_n$ tends to $-n/2$, $R_n$ tends to $n/2$, $p_n$ tends to $1/2$ and $Z_n$ tends to $0$.
But then, I have to represent graphically (using a histogram) an estimate of the law $\dfrac{Z_{1000}}{\sqrt{1000}}$, and determine the asymptotic positions of $L_n$ and $R_n$. I really don't know how to estimate this law. Someone could help me for this point, just with little indications ? (sorry for my bad english) ?