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Let $z_1,\ldots,z_n$ be an i.i.d sample from a sub-gaussian distribution. Define the $n$-by-$n$ p.s.d matrix $C_n := \frac{1}{n}zz^T + \text{diag}(z_1^2,\ldots,z_n^2)$, where $z:=(z_1,\ldots,z_n)$.

Question

What are some anti-concentration inequalities for $C_n$?

By anti-concentration, I mean inequality of the form "$C_n \succeq cI_n$" w.h.p., or something similar.

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

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