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.