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.

http://books.google.com/books?id=mYn...page&q&f=false

You should avoid the three series theorem, because we don't even want to mention having a second moment here.