1 n

We prove a quantitative version of a Silverstein’s Theorem on the 4-th moment condition for convergence in probability of the norm of a random matrix. More pre-cisely, we show that for a random matrix with i.i.d. entries, satisfying certain natural conditions, its norm cannot be small. Let w be a real random variable with Ew = 0 and Ew2 = 1, and let wij, i, j ≥ 1 be its i.i.d. copies. For integers n and p = p(n) consider the p×n matrix Wn = {wij}i≤p, j≤n, and consider its sample covariance matrix Γn: