# Variance of Squared Sum of (weakly) Dependent Random Variables $Var (\sum X_i)^2$

I am trying to work out an example I came up with in which I have to compute the variance of a squared sum of dependent random variables: \begin{align*} W:=\mathbf{V} \left( \sum_{i=1}^n X_i \right)^2. \end{align*} To motivate a bit, we know that if the $$X_i$$ are iid then $$W \leq cn^2$$ for some constant $$c$$, see e.g. Variance of squared sum of i.i.d random variables. I suspect that something similar is true in the case the $$X_i$$ are only "weakly dependent". Let's take this to mean, in the present situation, that $$corr(X_i^2,X_j^2) \leq r^{|i-j|}, 0 for all $$i,j$$.

## Question

Assuming the above and that the $$X_i$$ have, say, uniformly bounded moments of order 2 and 4, $$E X_i^2 \leq k, E X_i^4 \leq k'$$ for all $$i$$, can we work out a bound of the form $$W \leq C n^2$$ where $$C$$ only depends on $$r,r',k, k'$$?

Remark: I've been trying to compute this for a few hours now, writing $$\mathbf{V} \left( \sum_{i=1}^n X_i \right)^2 = \sum_{i,j,k,l} Cov (X_iX_j, X_lX_k).$$ The problem here seems to bound the $$(i,j)$$ and $$(k,l)$$ pairwise dependencies, and if anyone knows how to do this I would appreciate the help.

Remark2: I did a similar thing for $$\mathbf{V} \left( \sum_{i=1}^n X_i \right)\leq Cn$$ where $$C$$ is roughly speaking proportional to the $$l^1$$ norm of $$r'^{|i|}$$ above, which sort of inspired this question.