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). 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

• $\mu_i, \, \sigma_i < \infty \, \forall i$
• $\lim_{k \rightarrow \infty} \frac{1}{k} \sum_i^k \mu_i = \mu$, $\lim_{k \rightarrow \infty} \frac{1}{k} \sum_i^k \sigma^2_i = \sigma^2$
• $X_i$ has fast decaying tails, i.e. $P_X(X_i=x) = o(x^{-n}) \, \forall n \in \mathbb{N}$ for $x \rightarrow \pm \infty$

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