Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data

Let $$Z=[z_1, \dots z_n]$$ be a $$d \times n$$ matrix, where the $$z_i$$'s are iid random vactors with mean $$\mu \in \mathbb{R}^d$$ and $$d \times d$$ (population) covariance matrix $$\Sigma$$, but the entries $$z_{ij}'s$$ are not necessarily iid. Consider the (unscaled) sample covariance matrix $$C:= ZZ' \in \mathbb{R}^{d \times d}$$. I was wondering whether we have any results on the upper and lower bounds of $$\lambda_1(C), \lambda_d{(C)}$$ as a function of $$n$$ and $$d$$. I'm not assuming anything like the asyptotic regime in random matrix theory, so for example, no assumption like $$n, d \to \infty, d/n \to c$$. I'd rather have a non-asymtotic result, showing the dependence on $$d, n$$. Thank you and references would be really helpful!

If the random vectors are isotropic (meaning $$E[z_i z_i^T]=I$$), you can use the lower bound derived by Pavel Yaskov (2014):
• Thanks, this seems to be interesting for tight lower bounds. Vershynin's book "High Dimensional Probability with Application to data Science" gives some upper and lower bounds in the order of $\sqrt n +/- C \sqrt d$ (Theorem 4.6.1, P.98) , but assumes itotropic again. But I'll check this paper too... I'm just not sure how realistic this isotropic assumtion on $z_i's$ are, for real data the "features" $z_{ij}, z_{ik}$ are going to be correlated. Mar 17 '20 at 22:39
• okay, let me try to understand this. Say in my sample is $\{z_1,...z_n\}, z_i's$, are iid. Now if I assume that: each $z_i$ isotropic, then the polpulation covariance $cov(z_i) = \sigma^2 I_d$ (right?). This means (1) $cov(z_{ij}, z_{ik}) = 0 \forall j \ne k$. This means the $d$ features/covariates for each $z_i$ are pairwise uncorrelated (but not pairwise independent, except if they're normal), and also that (2) $Var(z_{ij}) = \sigma^2 \forall j ,$ so the all the individual features for each $z_i$ have same variance. Isn't it unrealistic to have uncorrelated features with same variances? Mar 18 '20 at 11:31