I'm wondering if there are any known results about minimum eigenvalue of matrices with i.i.d. heavy tailed columns. In particular, Theorem 5.62 of Roman Vershynin's notes (http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf) gives a bound on expected deviation of minimum singular value of a matrix with i.i.d. but Heavy-tailed columns. I was wondering if a high-probability analogue is known. For example, Theorem 5.58 of Vershynin's notes proves a result for i.i.d. sub-Gaussian columns. Was wondering if something similar is true for i.i.d. sub-exponential columns. Note that I only care about the minimum eigenvalue. I'm aware that for heavier tails the maximum eigenvalue is probably not going to be too concentrated. Intuitively, the more heavy-tailed the distribution is should only help in lower-bounding the minimum eigenvalue. While I'm aware of many results for concentration of minimum eigenvalues with heavy tailed i.i.d. rows (equivalent to concentration of sample covariance matrix) to my surprise I can't seem to find many for heavy tailed i.i.d. columns (concentration of Gram matrices).
(Note I am assuming that the number of rows is larger than the number of columns so I can not just transpose the matrix and use results for i.i.d. rows)
