## Concentration of $X^T\eta\eta^TX \in \mathbb R^d$ for i.i.d $(x_i,\eta_i)$ and sub-gaussian $\eta_i$

Suppose $$(x_1,\eta_1),\ldots,(x_n,\eta_n)$$ are $$n$$ i.i.d points in $$\mathbb R^{d+1}$$ such that $$\eta_1,\ldots,\eta_n$$ are $$\sigma$$-subgaussian. Let $$X \in \mathbb R^{n \times d}$$ be the vertical stacking of the $$x_i$$‘s and $$\eta \in \mathbb R^n$$ be the vertical stacking of the $$\eta_i$$‘s

# Question

Are there any concentration inequalities which can be liveraged to bound the matrix $$X^T\eta\eta^TX \in \mathbb R^{d \times d}$$ ?

# Observations

Naively, I’d guess that $$X^T\eta\eta^TX \preceq \sigma^2X^TX + \text{“small thing”}$$, with high probability.