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!



Bound on eigenvalues of sample covariance matrices in terms of $d, n$, where $n=$ sample size, $d=$ dimension of data
If the random vectors are *isotropic* (meaning $E[z\_i z\_i^T]=I$), you can use the lower bound derived by [Pavel Yaskov](https://projecteuclid.org/euclid.ecp/1465316785) (2014):



> 
> **Abstract:** We provide tight lower bounds on the smallest eigenvalue of a sample
>  covariance matrix of a centred isotropic random vector under weak or
>  no assumptions on its components.
> 
> 
> 


