I guess, it's tough one. I think I should use strong law of large numbers, but I don't know how.

Random variables $\displaystyle X_{1},X_{2},...$ are independent and $\displaystyle X_{k}$ with gaussian distribution $\displaystyle N(k^{1/2},k)$ k=1,2,..

Prove that following sequence

$\displaystyle \frac{1}{n^{2}}(X_{1}X_{2}+X_{3}X_{4}+...+X_{2n-1}X_{2n})$

is convergent in probability and find the limit.

Thx for any help.