Bounding the Smallest Singular Value of a Random Matrix Without Concentration

Given a random vector X in Rⁿ, set X₁,·, X_N to be independent copies of X and let Γ = 1/√N∑_i=1^N (X_i, ) e_i be the matrix whose rows are X₁√ N,·, X_N√N. We obtain new probabilistic lower bounds on the smallest singular value λ_min(Γ) in a rather general situation, and in particular, under the assumption that X is an isotropic random vector for which ⊃_t\in S^n-1P t, X|u L/u²⁺η for some L,η >0. Our results imply that a Bai-Yin-type lower bound holds for η >2, and, up to a log-factor, for η =2 as well. The bounds hold without any additional assumptions on the Euclidean norm |X|ℓ₂ⁿ. Moreover, we establish a nontrivial lower bound even without any higher moment assumptions (corresponding to the case η =0), if the linear forms satisfy a weak "small-ball" property. These estimates follow from general lower bounds on the infimum of the quadratic empirical process f →N^-1∑_i=1^N f²(X_i) which are of independent interest.