# Thread: random signs

1. ## random signs

Let $(X_n)_{n \geq 0}$ be i.i.d random signs, $P(X_n=1)=P(X_{n-1}=-1)=1/2$. Let $(a_n)_{n \geq 0}$ be a sequence of real numbers such that $\Sigma _{n=1} ^{\infty}a_n ^2 < \infty$. Show that $\Sigma _{n=1} ^ \infty a_n X_n$ is almost sure convergent.

By $\Sigma _{n=1} ^{\infty}a_n ^2 < \infty$, $a_n \rightarrow 0$ as $n \rightarrow \infty$. but i dont know how to go from here.

2. Have you seen the Khintchine-Kolmogorov Convergence Theorem?
It follows directly from that.
I found it online RIGHT out of Chow-Teicher.