Let $\{X_i\}_{i=1}^n$ be random vectors in $\mathbb{R}^d$ such that $$\mathbb{Var}{\sum_{i=1}^n X_i}\sim O(n)$$(I am not saying $\{X_i\}_{i=1}^n$ are iid). Can we say anything about $$\mathbb{Var}{\sum_{i=1}^n X_iX_i^T}$$, where I am defining $\mathbb{Var}{\sum_{i=1}^n X_iX_i^T} = \mathbf{E}[\sum_{i=1}^n X_iX_i^T]^2 - (\mathbf{E}[\sum_{i=1}^n X_iX_i^T])^2$.
I sort of hope to say it is of the same order. $X_i$ s are unit norm bounded.
EDIT : I think for the outer product the variance would be at most square of the variance of the sum of vectors