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][1]).
Called $\lambda = \sqrt{\sum_{i=1}^k \left( \frac{\mu_i}{\sigma_i} \right)^2}$ Let now assume that 
$$\lim_{k \rightarrow \infty} \frac{\lambda}{\sqrt{k}} = L$$, with L being a finite constant, i.e. $0<L<\infty$.

**Questions**

1) how does the mean and variance of the distribution scale with k in the limit of $k \rightarrow \infty$?

2) If now we assume generic non Gaussian r.v. $X_i$, such that 
- $\mu_i, \, \sigma_i < \infty \, \forall i$
- $\lim_{k \rightarrow \infty} \frac{1}{k} \sum_i^k \mu_i = \mu$, $\lim_{k \rightarrow \infty} \frac{1}{k} \sum_i^k \sigma^2_i =  \sigma^2$

how does the mean and variance of the distribution scale with k in the limit of $k \rightarrow \infty$ ?




  [1]: https://en.wikipedia.org/wiki/Noncentral_chi_distribution