Define $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ and independents.

It is known that $Y$ is distributed as a non-central chi (Noncentral chi distribution). Let $\lambda = \sqrt{\sum_{i=1}^k \left( \frac{\mu_i}{\sigma_i} \right)^2}.$ Now assume that $$\lim_{k \rightarrow \infty} \frac \lambda {\sqrt k} = L,$$ with $L$ being a finite constant, i.e. $0<L<\infty$.

Question

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