# Variance of linear combination

This is a follow up question to this.

Let $$(X_1,\ldots, X_n)$$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $$\mathcal{N}(0,1)$$ is a standard-normal distribution (so $$\mathcal{N}^2$$ is a chi-square distribution), and $$\alpha>0$$. Note that the RHS is a sum of scaled chi-square distributions.

I'm interested in the variance of a linear combination of the $$X_i$$'s, that is:

$$Var(\sum_{i=1}^{n}\alpha_iX_i)$$

where $$\alpha_i\in\mathbb{R}\setminus\lbrace0\rbrace$$.

If not much can be said about the exact variance, what can be said about an upper bound?