1. ## convergence of iid

I guess I can use LLN...

Show that for any sequence $X_1,\dots,X_n,\dots$ of iid random variables, $\frac{1}{n^2}\sum_{i=1}^n iX_i\to \frac{m}{2} \;\;a.s.$ if and only if $X_1$ is integrable and $\mathbb{E}[X_1]=m$.

Thanks.

2. First center the random variables and then use Kolmogorov's Three Series and then Kroncker's lemma.
You can actually quote the literature.
http://www.informaworld.com/smpp/con...6424572&db=all
I proved this type of result in my dissertation for non-independent stochastically dominated rvs a long time ago.

I found the exact result (exact is a pun by the way).
See Corollary 3 on page 128 from Chow and Teicher, which follows from Theorem 3.