My knowledge of probability is fairly limited. I need to evaluate
$\displaystyle Y = \sum_{i=1}^{n^4}{\left(\sum_{j=1}^{n^2}{X_j}\right)^2} + \displaystyle\sum_{i=1}^{n^4}{\left(\sum_{j=1}^{n^2}{X_j}\right)^2} + \displaystyle\sum_{i=1}^{n^4}{\left(\sum_{j=1}^{n^2}{X_j}\right)^2}$
Where $X_j \sim \mathcal{U}(0,1)$ are uniformly distributed random variables.
I'm interested in the limit for large $n$, so for my purposes each sum $\sum_{j}{X_j}$ is approximated by a Gaussian distribution with mean $n^2/2$ and variance $n^2 / 12$. To my understanding, the sum of $N$ squared normal RVs $Z_i \sim \mathcal{N}(\mu,\sigma^2)$ obeys
$\displaystyle W = \frac{1}{\sigma^2} \sum_{i=1}^{N}{Z_i} \sim \chi_{k}^2(\lambda)$
where $\chi_{k}^2$ is a noncentral chi-squared distribution with $k=N$ degrees of freedom and noncentrality parameter $\lambda = N\mu/\sigma$.
In my case, $N=n^4$, $k=n^4$, $\lambda = 6n^4$.
But now I have to add
$Y = \frac{1}{\sigma^2} W + \frac{1}{\sigma^2} W + \frac{1}{\sigma^2} W$
And I'm stuck.