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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 $$ corr(X_i^2,X_j^2) \leq r^{|i-j|}, 0<r<1\\ corr(X_i,X_j) \leq r'^{|i-j|}, 0<r'<1 $$ 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.

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