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.

No abstract

We prove the Marchenko-Pastur theorem for random matrices with i.i.d. rows and a general dependence structure within the rows by a simple modification of the standard Cauchy-Stieltjes resolvent method.

We show that a weak concentration property for quadratic forms of isotropic random vectors x is necessary and sufficient for the validity of the Marchenko-Pastur theorem for sample covariance matrices of random vectors having the form Cx, where C is any rectangular matrix with orthonormal rows. We also obtain some general conditions guaranteeing the weak concentration property.