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:


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?


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