Defined $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ it is known that Y is distributed as a non-central chi (Noncentral chi distribution); how does the mean and variance of the distribution scale with k in the limit of $k \rightarrow \infty$?