How do I prove that the weak law of large numbers holds?

Question

We have given $X_1,X_2,...$ an i.i.d. sequence of random variables with $$\Bbb{P}(X_1=1)=\Bbb{P}(X_1=-1)=\frac{1}{2}$$ From class we know that then the characteristic function is $\Phi_{X_i}(t)=\cos(t)$ for all $t\in \Bbb{R}$ and we know that $\lim_{n\rightarrow \infty} \cos^n\left(\frac{t}{n}\right)=1$ for all $t\in \Bbb{R}$. Now I need to use these two to show that $$\frac{X_1+...+X_n}{n}\rightarrow 0$$in probability.

I have unfortunately no idea where to start and how to do this using the two points I listed above. Could maybe someone give me a hint?

Thanks for you help.

Answer 1

The relevant theorems here are:

1. $X_n \Rightarrow X$ (convergence in distribution AKA weak convergence) if and only if the characteristic function of $X_n$ converges (pointwise) to the characteristic function of $X$. This is called Lévy's Continuity Theorem.

2. For a constant $c \in \mathbf{R}$, $X_n \Rightarrow c$ if and only if $X_n \xrightarrow{\mathbf P} c$. Usually convergence in probability is stronger but when the target is a constant the two modes of convergence are the same.